The Database of Integer Sequences, Part 9 

     Part of the On-Line Encyclopedia of Integer Sequences


This is a section of the main database for the On-Line Encyclopedia of Integer Sequences.

For more information see the following pages:
( www.research.att.com/~njas/sequences/ then )

    Seis.html:          Welcome
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    Spuzzle.html:       Puzzles 
    Shot.html:          Hot 
    classic.html:       Classics
    ol.html:            Superseeker 
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    home page:          www.research.att.com/~njas/
    


(start)



%I A127474
%S A127474 1,2,1,3,0,4,4,3,0,4,5,0,0,0,16,6,3,8,0,0,4,7,0,0,0,0,0,36,8,7,0,12,0,0,
%T A127474 0,16,9,0,16,0,0,0,0,0,36,10,5,0,0,32,0,0,0,0,16
%N A127474 Triangle, square of A054522.
%C A127474 Right border = A127473, squares of phi(n) terms. Row sums = A057660: (1, 3, 7, 11, 21,...)
%F A127474 (A054522)^2 as an infinite lower triangular matrix.
%e A127474 First few rows of the triangle are:
%e A127474 1;
%e A127474 2, 1;
%e A127474 3, 0, 4;
%e A127474 4, 3, 0, 4;
%e A127474 5, 0, 0, 0, 16;
%e A127474 6, 3, 8, 0, 0, 4;
%e A127474 7, 0, 0, 0, 0, 0, 36;
%e A127474 8, 7, 0, 12, 0, 0, 0, 16;
%e A127474 ...
%Y A127474 Cf. A127473, A057660, A054522.
%Y A127474 Adjacent sequences: A127471 A127472 A127473 this_sequence A127475 A127476 A127477
%Y A127474 Sequence in context: A030109 A058208 A070817 this_sequence A078024 A112469 A117778
%K A127474 nonn,tabl
%O A127474 1,2
%A A127474 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 15 2007

%I A078024
%S A078024 1,1,2,1,3,0,5,3,10,11,23,32,57,87,146,231,379,608,989,1595,2586,4179,6767,
%T A078024 10944,17713,28655,46370,75023,121395,196416,317813,514227,832042,1346267,
%U A078024 2178311,3524576,5702889,9227463,14930354,24157815,39088171,63245984,102334157
%V A078024 1,-1,2,-1,3,0,5,3,10,11,23,32,57,87,146,231,379,608,989,1595,2586,4179,6767,
%W A078024 10944,17713,28655,46370,75023,121395,196416,317813,514227,832042,1346267,
%X A078024 2178311,3524576,5702889,9227463,14930354,24157815,39088171,63245984,102334157
%N A078024 Expansion of (1-x)/(1-2*x^2-x^3).
%F A078024 Fibonacci(n+2) - Lucas(n) + 2(-1)^n.
%Y A078024 Cf. A008346.
%Y A078024 Adjacent sequences: A078021 A078022 A078023 this_sequence A078025 A078026 A078027
%Y A078024 Sequence in context: A058208 A070817 A127474 this_sequence A112469 A117778 A072127
%K A078024 sign
%O A078024 0,3
%A A078024 njas, Nov 17 2002

%I A112469
%S A112469 1,1,2,1,3,0,5,3,10,11,23,32,57,87,146,231,379,608,989,1595,2586,4179,6767,
%T A112469 10944,17713,28655,46370,75023,121395,196416,317813,514227,832042,1346267,
%U A112469 2178311,3524576,5702889,9227463,14930354,24157815,39088171,63245984,102334157
%V A112469 1,1,2,1,3,0,5,-3,10,-11,23,-32,57,-87,146,-231,379,-608,989,-1595,2586,-4179,6767,
%W A112469 -10944,17713,-28655,46370,-75023,121395,-196416,317813,-514227,832042,-1346267,
%X A112469 2178311,-3524576,5702889,-9227463,14930354,-24157815,39088171,-63245984,102334157
%N A112469 Partial sums of (-1)^n*F(n-1).
%C A112469 Diagonal sums of Riordan array (1/(1-x),x/(1+x)), A112468.
%F A112469 G.f.: (1+x)/((1-x)(1+x-x^2)); a(n)=sum{k=0..floor(n/2), sum{j=0..n-2k, C(n-k-j-1, n-2k-j)*(-1)^(n-j)}.
%p A112469 a[0]:=1:a[1]:=1:a[2]:=2:a[3]:=1:for n from 4 to 50 do a[n]:=2*a[n-2]-a[n-3] od: seq(a[n], n=0..42); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 04 2008
%Y A112469 Cf. A078024.
%Y A112469 Adjacent sequences: A112466 A112467 A112468 this_sequence A112470 A112471 A112472
%Y A112469 Sequence in context: A070817 A127474 A078024 this_sequence A117778 A072127 A119805
%K A112469 easy,sign
%O A112469 0,3
%A A112469 Paul Barry (pbarry(AT)wit.ie), Sep 06 2005

%I A117778
%S A117778 2,1,3,0,10,0,29,0,97,0,293,0
%N A117778 Total number of palindromic primes in base 4 with n digits.
%C A117778 Every palindrome with an even number of digits is divisible by 11 (in base 4) and therefore is composite (not prime). Hence there is only one palindromic prime with an even number of digits.
%H A117778 Eric Weisstein: <a href="http://mathworld.wolfram.com/PalindromicPrime.html">Palindromic Prime</a>.
%Y A117778 Cf. A029972, A117699.
%Y A117778 Adjacent sequences: A117775 A117776 A117777 this_sequence A117779 A117780 A117781
%Y A117778 Sequence in context: A127474 A078024 A112469 this_sequence A072127 A119805 A111957
%K A117778 nonn
%O A117778 1,1
%A A117778 Martin Renner (martin.renner(AT)gmx.net), Apr 15 2006

%I A072127
%S A072127 1,1,1,2,1,3,0,17,0,20,1,52,1,0,0,204,0,24,0
%N A072127 Number of equivalence classes of pairs of complex Golay sequences of length n.
%D A072127 R. Craigen, W. Holzmann and H. Kharaghani, Complex Golay sequences: structure and applications, Discrete Math., 252 (2002), 73-89.
%Y A072127 Cf. A072128.
%Y A072127 Adjacent sequences: A072124 A072125 A072126 this_sequence A072128 A072129 A072130
%Y A072127 Sequence in context: A078024 A112469 A117778 this_sequence A119805 A111957 A125168
%K A072127 nonn,nice
%O A072127 1,4
%A A072127 njas, Jun 26 2002
%E A072127 The number at length 20 is large but unknown. There are none at length 21.

%I A119805
%S A119805 1,1,2,1,3,1,1,0,5,1,1,0,1,0,0,0,8,1,1,0,1,0,0,1,0,0,0,0,0,0,0,0,12,1,1,
%T A119805 0,1,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,1,1,0,1,0,
%U A119805 0,1,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A119805 a(1) = 1. For m >= 0 and 1 <= k <= 2^m, a(2^m +k) = number of earlier terms of the sequence which equal k.
%e A119805 8 = 2^2 + 4; so for a(8) we want the number of terms among terms a(1), a(2),... a(7) which equal 4. So a(8) = 0.
%o A119805 (PARI) A119805(mmax)= { local(a,ncopr); a=[1]; for(m=0,mmax, for(k=1,2^m, ncopr=0; for(i=1,2^m+k-1, if( a[i]==k, ncopr++; ); ); a=concat(a,ncopr); ); ); return(a); } { print(A119805(6)); } - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 30 2006
%Y A119805 Cf. A119804.
%Y A119805 Adjacent sequences: A119802 A119803 A119804 this_sequence A119806 A119807 A119808
%Y A119805 Sequence in context: A112469 A117778 A072127 this_sequence A111957 A125168 A051794
%K A119805 easy,nonn
%O A119805 1,3
%A A119805 Leroy Quet (qq-quet(AT)mindspring.com), May 24 2006
%E A119805 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 30 2006

%I A111957
%S A111957 1,1,1,1,1,2,1,3,1,1,1,1,1,1,1,1,1,4,1,1,2,1,1,1,1,1,1,1,1,3,1,7,1,3,1,
%T A111957 1,1,1,2,1,1,2,1,1,2,1,1,1,1,11,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,1,
%U A111957 1,18,1,1,4,3,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,29,1,1,1,1,1,1
%N A111957 Triangle read by rows: T(n,k) = gcd(Fibonacci(n), Lucas(k)), 1 <= k <= n.
%D A111957 P. Ribenboim, FFF (Favorite Fibonacci Flowers), Fib. Q. 43 (No. 1, 2005), 3-14.
%F A111957 T(n, k) = Lucas(g), where g = gcd(n, k), if n/g is even; = 2 if n/g is odd and 3|g; = 1 otherwise.
%Y A111957 Cf, A000045, A000032, A111946, A111956.
%Y A111957 Adjacent sequences: A111954 A111955 A111956 this_sequence A111958 A111959 A111960
%Y A111957 Sequence in context: A117778 A072127 A119805 this_sequence A125168 A051794 A110969
%K A111957 nonn,tabl
%O A111957 1,6
%A A111957 njas, Nov 28 2005

%I A125168
%S A125168 1,1,1,2,1,3,1,1,1,1,1,1,1,1,3,4,1,1,1,5,3,1,1,1,1,1,3,1,1,1,1,1,3,1,1,
%T A125168 4,1,1,3,1,1,7,1,1,5,1,1,3,1,5,3,1,1,1,1,7,3,1,1,1,1,1,1,2,1,1,1,1,3,7,
%U A125168 1,1,1,1,5,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,3,1,1,1,1,1,1,4,1,1,1,1,7
%N A125168 a(n)=gcd(n,b(n)) where b(n)=the number of proper divisors of n.
%C A125168 First occurrence of k: 1, 4, 6, 16, 20, 3240000, 42, 256, 162, 18662400, 132, 5308416, 832, 784, 120, 65536, 612, 2985984, 912, 1600, 9240, 98010000, 1380, 1296, 100800, ..., (10^7). - Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 23 2007
%C A125168 Do all values appear? - Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 23 2007
%e A125168 a(6)=3 because 6 has 3 proper divisors {1,2,3} and the gcd(6,3) is 3.
%t A125168 f[n_] := GCD[n, DivisorSigma[0, n] - 1]; Array[f, 105] (* Robert G. Wilson v *).
%Y A125168 Cf. A032741, A009191.
%Y A125168 Adjacent sequences: A125165 A125166 A125167 this_sequence A125169 A125170 A125171
%Y A125168 Sequence in context: A072127 A119805 A111957 this_sequence A051794 A110969 A006083
%K A125168 easy,nonn
%O A125168 1,4
%A A125168 Mitch Cervinka (puritan(AT)toast.net), Jan 12 2007
%E A125168 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 23 2007

%I A051794
%S A051794 1,1,1,1,1,1,0,1,1,1,1,1,1,0,1,1,1,1,1,1,2,1,3,1,1,1,1,2,5,3,7,1,
%T A051794 1,1,0,5,13,7,15,1,1,0,5,13,33,15,31,1,2,5,23,33,81,31,63,2,9,
%U A051794 23,79,81,193,63,128,9,41,79,239,193,449,128,265,41,161,239
%V A051794 1,1,1,1,1,1,0,1,1,1,1,1,1,0,-1,1,1,1,1,1,2,-1,-3,1,1,1,1,2,5,-3,-7,1,
%W A051794 1,1,0,5,13,-7,-15,1,1,0,-5,13,33,-15,-31,1,2,-5,-23,33,81,-31,-63,2,9,
%X A051794 -23,-79,81,193,-63,-128,9,41,-79,-239,193,449,-128,-265,41,161,-239
%N A051794 a(n)=sum((-1)^i*a(i),i=n-6..n-1), a(1)=1,a(2)=1,a(3)=1,a(4)=1,a(5)=1,a(6)=1.
%Y A051794 Adjacent sequences: A051791 A051792 A051793 this_sequence A051795 A051796 A051797
%Y A051794 Sequence in context: A119805 A111957 A125168 this_sequence A110969 A006083 A080301
%K A051794 easy,nice,sign
%O A051794 1,21
%A A051794 Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 10 1999

%I A110969
%S A110969 1,1,1,1,2,1,3,1,1,1,1,4,3,1,3,1,3,5,1,2,2,3,1,5,1,1,5,7,3,1,3,1,3,7,3,
%T A110969 1,2,5,1,9
%N A110969 Length of the runs of ones in A014963.
%C A110969 Unbounded sequence.
%e A110969 a(5)=2 because the fifth run of ones in A014963 is of length 2.
%Y A110969 Cf. A014963.
%Y A110969 Adjacent sequences: A110966 A110967 A110968 this_sequence A110970 A110971 A110972
%Y A110969 Sequence in context: A111957 A125168 A051794 this_sequence A006083 A080301 A057021
%K A110969 nonn
%O A110969 1,5
%A A110969 Franz Vrabec (franz.vrabec(AT)planetuniqa.at), Sep 27 2005

%I A006083 M0125
%S A006083 1,2,1,3,1,1,1,3,3,3,1,3,1,3,5,3,1,5,1,3,7,3,1,7,1,3,9,3,1,9,1,3,11,3,
%T A006083 1,11,1,3,13,3,1,13,1,3,15,3,1,15,1,3,17,3,1,17,1,3,19,3,1,19,1,3,21,3,
%U A006083 1,21,1,3,23,3,1,23,1,3,25,3,1,25,1,3,27,3,1,27,1,3,29,3,1,29,1,3,31,3
%N A006083 Continued fraction for e/2.
%D A006083 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 601.
%H A006083 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%H A006083 <a href="http://www.research.att.com/~njas/sequences/Sindx_Con.html#confC">Index entries for continued fractions for constants</a>
%F A006083 a(1)=1, a(2)=2, a(3)=1, a(4)=3, a(5)=1, a(6)=1, a(7)=1, a(8)=3, then for k>=1 a(6k+3)=a(6k+6)=2k+1, a(6k+4)=a(6k+8)=3, a(6k+5)=a(6k+7)=1. - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 08 2003
%Y A006083 Adjacent sequences: A006080 A006081 A006082 this_sequence A006084 A006085 A006086
%Y A006083 Sequence in context: A125168 A051794 A110969 this_sequence A080301 A057021 A119804
%K A006083 cofr,nonn,easy
%O A006083 1,2
%A A006083 njas, Jeffrey Shallit
%E A006083 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003

%I A080301
%S A080301 0,1,0,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T A080301 1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,2,1,3,1,1,1,4,1,1,1,1
%V A080301 0,-1,0,-1,-1,-1,-1,-1,-1,-1,0,-1,1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,
%W A080301 -1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,-1,1,-1,-1,-1,-1,-1,2,-1,3,-1,-1,-1,4,-1,-1,-1,
%X A080301 -1
%N A080301 Local ranking function for totally balanced binary sequences: if n's binary expansion is totally balanced (A080116(n)=1), then a(n) is its zero-based position among A000108((A000523(n)+1)/2) lexicographically ordered totally balanced binary sequences of the same width, otherwise -1.
%C A080301 Maple procedure CatalanRank is adapted from the algorithm 3.23 of the CAGES book.
%H A080301 D. L. Kreher and D. R. Stinson, <a href="http://www.math.mtu.edu/~kreher/cages.html">Combinatorial Algorithms, Generation, Enumeration and Search</a>, CRC Press, 1998.
%e A080301 We have Cat(0)=1 totally balanced binary sequences of length 2*0: 0, thus a(0)=0, Cat(1)=1 of length 2*1: 10, thus a(2)=0, Cat(2)=2 of length 2*2: 1010 (= 10.) and 1100 (= 12.), thus a(10)=0 and a(12)=1, plus altogether Cat(3)=5 totally balanced binary sequences of length 2*3: 101010 (= 42), 101100 (= 44), 110010 (= 50), 110100 (= 52), 111000 (= 56), thus a(42)=0, a(44)=1, a(50)=2, a(52)=3 and a(56)=4. Et cetera.
%p A080301 A080301 := n -> `if`(0 = A080116(n),-1,CatalanRank((A000523(n)+1)/2,n));
%p A080301 CatalanRank := proc(n,aa) local y,r,lo,a; a := aa; r := 0; y := -1; lo := 0; while (a > 0) do if(0 = (a mod 2)) then r := r+1; lo := lo + A009766(r,y); else y := y+1; fi; a := floor(a/2); od; RETURN((binomial(2*n,n)/(n+1))-(lo+1)); end;
%Y A080301 Used to compute A080300. Cf. A009766, A000523.
%Y A080301 Adjacent sequences: A080298 A080299 A080300 this_sequence A080302 A080303 A080304
%Y A080301 Sequence in context: A051794 A110969 A006083 this_sequence A057021 A119804 A058564
%K A080301 sign
%O A080301 0,51
%A A080301 Antti Karttunen (my_firstname.my_surname(AT)iki.fi) Feb 21 2003

%I A057021
%S A057021 1,2,1,3,1,1,1,4,3,2,1,3,1,1,1,5,1,2,1,1,1,1,1,2,3,2,1,3,1,1,1,2,1,2,1,
%T A057021 9,1,1,1,4,1,1,1,1,1,1,1,5,1,2,1,3,1,1,1,1,1,2,1,1,1,1,3,7,1,1,1,1,1,1
%N A057021 Denominator of (sum of factors of n / number of factors of n).
%C A057021 a(n) = 1 when n is listed in A003601, a(n) > 1 when n is listed in A049642 - Alonso Delarte (alonso.delarte(AT)gmail.com), Jan 31 2006
%F A057021 a(n) =A057020(n)*A000005(n)/A000203(n) =A000005(n)/A009205(n)
%e A057021 a(12)=3 since the 6 factors of 12 are 1, 2, 3, 4, 6 and 12, and 1+2+3+4+6+12=28 and 28/6=14/3
%t A057021 Denominator[Table[(Plus @@ Divisors[n])/Length[Divisors[n]], {n, 70}]] (Alonso Delarte (alonso.delarte(AT)gmail.com))
%Y A057021 Cf. A000005, A000203, A009205, A054025, A057020 (numerator), A057022.
%Y A057021 Adjacent sequences: A057018 A057019 A057020 this_sequence A057022 A057023 A057024
%Y A057021 Sequence in context: A110969 A006083 A080301 this_sequence A119804 A058564 A087157
%K A057021 frac,nonn
%O A057021 1,2
%A A057021 Henry Bottomley (se16(AT)btinternet.com), Jul 21 2000

%I A119804
%S A119804 0,1,1,2,1,3,1,1,1,6,1,1,0,0,1,0,4,9,1,1,1,0,1,0,0,1,0,0,0,0,0,0,13,14,
%T A119804 1,1,1,0,1,0,0,1,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,36,21,1,1,
%U A119804 1,0,1,0,0,1,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0
%N A119804 a(0) = 0. For m >= 0 and 0 <= k <= 2^m -1, a(2^m +k) = number of earlier terms of the sequence which equal k.
%e A119804 8 = 2^3 + 0; so for a(8) we want the number of terms among terms a(1), a(2),... a(7) which equal 0. So a(8) = 1.
%o A119804 (PARI) A119804(mmax)= { local(a,ncopr); a=[0]; for(m=0,mmax, for(k=0,2^m-1, ncopr=0; for(i=1,2^m+k, if( a[i]==k, ncopr++; ); ); a=concat(a,ncopr); ); ); return(a); } { print(A119804(6)); } - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 30 2006
%Y A119804 Cf. A119805.
%Y A119804 Adjacent sequences: A119801 A119802 A119803 this_sequence A119805 A119806 A119807
%Y A119804 Sequence in context: A006083 A080301 A057021 this_sequence A058564 A087157 A123507
%K A119804 easy,nonn
%O A119804 0,4
%A A119804 Leroy Quet (qq-quet(AT)mindspring.com), May 24 2006
%E A119804 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 30 2006

%I A058564
%S A058564 1,0,1,1,1,2,1,3,1,1,2,0,1,2,4,1,3,4,3,3,2,10,2,6,7
%V A058564 1,0,-1,-1,1,2,-1,3,-1,-1,-2,0,1,-2,4,-1,-3,-4,3,3,-2,10,-2,-6,-7
%N A058564 McKay-Thompson series of class 21B for Monster.
%D A058564 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%e A058564 T21B = 1/q - q - q^2 + q^3 + 2*q^4 - q^5 + 3*q^6 - q^7 - q^8 - 2*q^9 + q^11 - ...
%Y A058564 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%Y A058564 Adjacent sequences: A058561 A058562 A058563 this_sequence A058565 A058566 A058567
%Y A058564 Sequence in context: A080301 A057021 A119804 this_sequence A087157 A123507 A122580
%K A058564 sign
%O A058564 -1,6
%A A058564 njas, Nov 27, 2000

%I A087157
%S A087157 1,2,1,3,1,1,2,1,4,1,1,1,2,1,2,1,3,1,1,2,1,5,1,1,1,1,2,1,2,1,2,1,3,1,1,
%T A087157 2,1,3,1,1,2,1,4,1,1,1,2,1,2,1,3,1,1,2,1,6,1,1,1,1,1,2,1,2,1,2,1,2,1,3,
%U A087157 1,1,2,1,3,1,1,2,1,3,1,1,2,1,4,1,1,1,2,1,2,1,3,1,1,2,1,4,1,1,1,2,1,2,1
%N A087157 Satisfies a(1)=1, a(A087158(n+1)) = a(n)+1, with a(m)=1 for all m not found in A087158, where A087158(n+2)=A087158(n+1)+a(n)+1.
%C A087157 Removing all the 1's results in the original sequence with every term incremented by 1.
%F A087157 Records are given by A055588(n): a(A055588(n))=n, where A055588(n)=Fibonacci(2n-2)+1, and Fibonacci(n)=A000045(n).
%e A087157 Initialize all terms to 1. Set a(1)=1, go one term forward,
%e A087157 set a(2)=a(1)+1=2, go 2 terms forward,
%e A087157 set a(4)=a(2)+1=3, go 3 terms forward,
%e A087157 set a(7)=a(3)+1=2, go 2 terms forward,
%e A087157 set a(9)=a(4)+1=4, go 4 terms forward,
%e A087157 set a(13)=a(5)+1=1, etc.
%e A087157 The indices 1,2,4,7,9,13,... form A087158.
%Y A087157 Cf. A087158, A085246.
%Y A087157 Adjacent sequences: A087154 A087155 A087156 this_sequence A087158 A087159 A087160
%Y A087157 Sequence in context: A057021 A119804 A058564 this_sequence A123507 A122580 A107041
%K A087157 nonn
%O A087157 1,2
%A A087157 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 22 2003

%I A123507
%S A123507 1,2,1,3,1,1,2,2,3,2,4,3,5,5,5,7,8
%N A123507 Lengths of bit runs in A123506.
%C A123507 The sequence uses operations based on the second nontrivial Riemann zero: (1/2 + i*t), t = 21.022039639... A123504 and A123505 use the first nontrivial zero.
%D A123507 John Derbyshire, Prime Obsession, Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Plume - a Pengiun Group, NY, 2003, pgs. 198-9.
%F A123507 Record the numbers of consecutive bit runs of A123506.
%e A123507 a(4) = 3 since A123506 = 0, 1, 1, 0, 1, 1, 1...
%Y A123507 Cf. A123504, A123505, A123506, A100060, A102522, A102523.
%Y A123507 Adjacent sequences: A123504 A123505 A123506 this_sequence A123508 A123509 A123510
%Y A123507 Sequence in context: A119804 A058564 A087157 this_sequence A122580 A107041 A070099
%K A123507 nonn
%O A123507 2,2
%A A123507 Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 01 2006

%I A122580
%S A122580 1,2,1,3,1,1,2,3,2,3,3,1,5,4,0,5,3,0,7,8,3,9,6,2,9,10,3,13,11,1,15,13,3,
%T A122580 18,14,3,22,20,7,27,21,3,29,27,8,34,30,7,42,37,8,48,39,9,55,50,13,66,52,
%U A122580 11,74
%V A122580 1,-2,-1,3,-1,1,2,-3,-2,3,-3,-1,5,-4,0,5,-3,0,7,-8,-3,9,-6,-2,9,-10,-3,13,-11,-1,15,
%W A122580 -13,-3,18,-14,-3,22,-20,-7,27,-21,-3,29,-27,-8,34,-30,-7,42,-37,-8,48,-39,-9,55,-50,
%X A122580 -13,66,-52,-11,74
%N A122580 Number of partitions of n with crank congruent to 0 mod 3, minus number of partitions of n with crank congruent to 1 mod 3.
%C A122580 For a partition p, let l(p) = largest part of p, w(p) = number of 1's in p, m(p) = number of parts of p larger than w(p). The crank of p is given by l(p) if w(p) = 0, otherwise m(p)-w(p).
%H A122580 Daniel M. Kane, <a href="http://web.mit.edu/dankane/www/AndrewsConjecture.pdf">RESOLUTION OF A CONJECTURE OF ANDREWS AND LEWIS INVOLVING CRANKS OF PARTITIONS</a>
%F A122580 G.f.: Product((1-x^n)/(1+x^n+x^(2*n)),n=1..infinity). Euler transform of period 3 sequence [ -2,-2,-1, ...].
%Y A122580 Adjacent sequences: A122577 A122578 A122579 this_sequence A122581 A122582 A122583
%Y A122580 Sequence in context: A058564 A087157 A123507 this_sequence A107041 A070099 A126760
%K A122580 easy,sign
%O A122580 0,2
%A A122580 Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 20 2006

%I A107041
%S A107041 1,1,2,1,3,1,1,2,4,1,1,3,1,2,1,1,2,2,2,1,3,1,1,2,2,3,1,3,1,2,1,1,2,1,1,
%T A107041 2,1,3,1,1,2,2,2,1,4,1,2,1,1,2,1,1,2,4,1,1,2,2
%N A107041 First differences of indices of square-free Pell numbers.
%Y A107041 Cf. A000129, A107038, A107039.
%Y A107041 Adjacent sequences: A107038 A107039 A107040 this_sequence A107042 A107043 A107044
%Y A107041 Sequence in context: A087157 A123507 A122580 this_sequence A070099 A126760 A019442
%K A107041 nonn
%O A107041 0,3
%A A107041 Paul Barry (pbarry(AT)wit.ie), May 09 2005

%I A070099
%S A070099 0,0,1,0,1,0,1,1,1,1,2,1,3,1,1,2,4,1,4,2,2,2,5,1,4,2,4,3,6,2,6,3,4,3,5,
%T A070099 3,8,3,4,3,8,3,9,5,5,4,10,3,9,4,6,5,11,4,8,5,7,6,12,3,13,6,8,7,9,4,14,
%U A070099 7,8,5,15,5,15,7,9,8,13,6,16,6,11,8,17,5,13
%N A070099 Number of integer triangles with perimeter n and relatively prime side lengths which are acute and isosceles.
%H A070099 R. Zumkeller, <a href="http://www.research.att.com/~njas/sequences/a070080.txt">Integer-sided triangles</a>
%Y A070099 Cf. A070080, A070081, A070082, A070093, A059169, A051493, A070091, A070094, A070098, A070084, A070125.
%Y A070099 Adjacent sequences: A070096 A070097 A070098 this_sequence A070100 A070101 A070102
%Y A070099 Sequence in context: A123507 A122580 A107041 this_sequence A126760 A019442 A007740
%K A070099 nonn
%O A070099 1,11
%A A070099 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 05 2002

%I A126760
%S A126760 0,1,1,1,1,2,1,3,1,1,2,4,1,5,3,2,1,6,1,7,2,3,4,8,1,9,5,1,3,10,2,11,1,4,
%T A126760 6,12,1,13,7,5,2,14,3,15,4,2,8,16,1,17,9,6,5,18,1,19,3,7,10,20,2,21,11,
%U A126760 3,1,22,4,23,6,8,12,24,1,25,13,9,7,26,5,27,2,1,14,28,3,29,15,10,4,30,2
%N A126760 A126759(n) - 1.
%C A126760 For further information see A126759, which is the main entry for this sequence.
%Y A126760 Adjacent sequences: A126757 A126758 A126759 this_sequence A126761 A126762 A126763
%Y A126760 Sequence in context: A122580 A107041 A070099 this_sequence A019442 A007740 A117811
%K A126760 nonn
%O A126760 0,6
%A A126760 njas, Feb 19 2007

%I A019442
%S A019442 1,1,1,1,2,1,3,1,1,2,5,1,3,3,2,2,8,1,9,2,3,5,11,1,10,3,1,3,14,2,15,4,5,8
%N A019442 Duplicate of A007740.
%Y A019442 Adjacent sequences: A019439 A019440 A019441 this_sequence A019443 A019444 A019445
%Y A019442 Sequence in context: A107041 A070099 A126760 this_sequence A007740 A117811 A051793
%K A019442 dead
%O A019442 1,5

%I A007740
%S A007740 1,1,1,1,2,1,3,1,1,2,5,1,3,3,2,2,8,1,9,2,3,5,11,1,10,3,1,3,14,2,15,4,5,8,
%T A007740 6,1,9,9,3,2,4,3,21,5,2,11,23,2,21,10,8,3,26,1,10,3,9,14,29,2,5,15,3,8,6,
%U A007740 5,11,8,11,6,35,1,6,9,10,9,15,3,39,2,1,4,41,3,8,21,14,5,44,2,3,11,15,23
%N A007740 Period of repeating digits of 1/n in base 9.
%C A007740 Not multiplicative. Smallest counterexample a(5) = a(16) = 2, but a(80) = 2. David W. Wilson (davidwwilson(AT)comcast.net) Jun 09, 2005.
%H A007740 <a href="http://www.research.att.com/~njas/sequences/Sindx_1.html#1overn">Index entries for sequences related to decimal expansion of 1/n</a>
%Y A007740 Adjacent sequences: A007737 A007738 A007739 this_sequence A007741 A007742 A007743
%Y A007740 Sequence in context: A070099 A126760 A019442 this_sequence A117811 A051793 A065371
%K A007740 nonn
%O A007740 1,5
%A A007740 njas, Hal Sampson [ hals(AT)easynet.com ]

%I A117811
%S A117811 1,1,2,1,3,1,1,2,5,1,6,1,7,1,2,2,1,3,10,1,11,1,3,2,13,1,14,1,15,1,1,
%T A117811 4,17,1,2,3,19,1,5,2,21,1,22,1,23,1,6,2,1,5,26,1,3,3,7,2,29,1,30,1,
%U A117811 31,1,2,4,33,1,34,1,35,1,1,6,37,1,38,1,39,1,10,2,41,1,42,1,43,1,11
%N A117811 Two-column table read by rows: row n (n >= 1) gives a squarefree integer x and an integer y such that n = x*y^2.
%e A117811 Table begins:
%e A117811 1 1
%e A117811 2 1
%e A117811 3 1
%e A117811 1 2 (so 4 = 1*2^2)
%e A117811 5 1
%e A117811 6 1
%e A117811 7 1
%e A117811 2 2 (so 8 = 2*2^2)
%e A117811 1 3
%e A117811 10 1
%e A117811 11 1
%e A117811 3 2
%e A117811 13 1
%e A117811 14 1
%e A117811 15 1
%e A117811 1 4
%o A117811 (MAGMA) at:=0; for n in [1..100] do print Squarefree(n); end for;
%Y A117811 The two columns are A007913, A000188.
%Y A117811 Adjacent sequences: A117808 A117809 A117810 this_sequence A117812 A117813 A117814
%Y A117811 Sequence in context: A126760 A019442 A007740 this_sequence A051793 A065371 A006346
%K A117811 nonn,tabf
%O A117811 1,3
%A A117811 njas, Dec 23 2006

%I A051793
%S A051793 1,1,1,1,0,1,1,1,1,0,1,1,1,1,2,1,3,1,1,2,5,3,7,1,0,5,13,7,15,0,
%T A051793 5,13,33,15,30,5,23,33,81,30,55,23,79,81,192,55,87,79,239,
%U A051793 192,439,87,95,239,670,439,965,95,49,670,1779,965,2025,49,768
%V A051793 1,1,1,1,0,1,1,1,1,0,-1,1,1,1,2,-1,-3,1,1,2,5,-3,-7,1,0,5,13,-7,-15,0,
%W A051793 -5,13,33,-15,-30,-5,-23,33,81,-30,-55,-23,-79,81,192,-55,-87,-79,-239,
%X A051793 192,439,-87,-95,-239,-670,439,965,-95,49,-670,-1779,965,2025,49,768
%N A051793 a(n)=sum((-1)^i*a(i),i=n-4..n-1), a(1)=1,a(2)=1,a(3)=1,a(4)=1.
%Y A051793 Adjacent sequences: A051790 A051791 A051792 this_sequence A051794 A051795 A051796
%Y A051793 Sequence in context: A019442 A007740 A117811 this_sequence A065371 A006346 A088742
%K A051793 easy,nice,sign
%O A051793 1,15
%A A051793 Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 10 1999

%I A065371
%S A065371 1,1,1,1,2,1,3,1,1,2,6,1,7,3,2,1,10,1,11,2,3,6,14,1,4,7,1,3,19,2,20,1,
%T A065371 6,10,6,1,25,11,7,2,28,3,29,6,2,14,32,1,9,4,10,7,37,1,12,3,11,19,42,2,
%U A065371 43,20,3,1,14,6,48,10,14,6,51,1,52,25,4,11,18,7,57,2,1,28,60,3,20,29
%N A065371 a(1) = 1, a(prime(i)) = prime(i) - i for i > 0 and a(u * v) = a(u) * a(v) for u, v > 0.
%C A065371 a(n) > 0 and a(n) < n for all n > 1.
%e A065371 a(210) = a(2*3*5*7) = a(2)*a(3)*a(5)*a(7) = (prime(1)-1)*(prime(2)-2)*(prime(3)-3)*(prime(4)-4) = (2-1)*(3-2)*(5-3)*(7-4) = 1*1*2*3 = 6.
%Y A065371 A000040, A014689, A065372, A065373, A065374.
%Y A065371 Adjacent sequences: A065368 A065369 A065370 this_sequence A065372 A065373 A065374
%Y A065371 Sequence in context: A007740 A117811 A051793 this_sequence A006346 A088742 A130296
%K A065371 mult,nonn
%O A065371 1,5
%A A065371 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Nov 01 2001

%I A006346 M0126
%S A006346 0,1,1,2,1,3,1,1,3,2,1,6,3,2,1,3,1,1,6,3,2,4,1,1,3,2,1,3,1,6,4,2,1,2,4,
%T A006346 3,1,8,3,2,1,6,3,2,1,3,1,1,6,3,2,4,1,1,3,2,1,3,1,30,6,3,2,4,1,1,3,2,1,
%U A006346 3,1,6,4,2,1,2,4,3,1,8,3,2,1,6,3,2,1,3,1,1,6,3,2,4,1,1,3,2,1,3,1,6,4,2
%N A006346 The Sally sequence: the length of repetition avoided in A006345.
%C A006346 Comment from T. D. Noe, Oct 14 2006: In the first 1000 terms, only patterns of lengths 0, 1, 2, 3, 4, 6, 8, 24, 30, and 108 are avoided.
%D A006346 N. S. Hellerstein, personal communication.
%H A006346 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b006346.txt">Table of n, a(n) for n=1..1000</a>
%H A006346 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a6345.html">Illustration of initial terms</a>
%H A006346 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SallySequence.html">Link to a section of The World of Mathematics.</a>
%Y A006346 Cf. A006345.
%Y A006346 Adjacent sequences: A006343 A006344 A006345 this_sequence A006347 A006348 A006349
%Y A006346 Sequence in context: A117811 A051793 A065371 this_sequence A088742 A130296 A126705
%K A006346 nonn,nice,easy
%O A006346 1,4
%A A006346 njas
%E A006346 More terms from Naohiro Nomoto (6284968128(AT)geocities.co.jp), May 21 2001

%I A088742
%S A088742 1,1,2,1,3,1,1,3,3,1,1,2,1,4,1,3,3,1,1,2,1,3,1,1,3,4,1,2,1,4,1,3,3,
%T A088742 1,1,2,1,3,1,1,3,3,1,1,2,1,4,1,3,4,1,2,1,3,1,1,3,4,1,2,1,4,1,3,3,1,
%U A088742 1,2,1,3,1,1,3,3,1,1,2,1,4,1,3,3,1,1,2,1,3,1,1,3,4,1,2,1,4,1,3,4,1
%N A088742 Run lengths of A088023.
%Y A088742 First differences of A091072.
%Y A088742 Cf. A088023, A088743, A088744.
%Y A088742 Adjacent sequences: A088739 A088740 A088741 this_sequence A088743 A088744 A088745
%Y A088742 Sequence in context: A051793 A065371 A006346 this_sequence A130296 A126705 A113924
%K A088742 nonn
%O A088742 1,3
%A A088742 Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 12 2003
%E A088742 More terms and better description from Ralf Stephan, Sep 03 2004

%I A130296
%S A130296 1,2,1,3,1,1,4,1,1,1,5,1,1,1,1,6,1,1,1,1,1,7,1,1,1,1,1,1
%N A130296 Triangle read by rows, reversals of A051340.
%C A130296 Row sums = (1, 3, 5,...). A130296^2 = A130297.
%F A130296 Reversal of A051340. By rows, "n" followed by (n-1) 1's. (1,2,3...) in the left border, all 1's in other columns.
%e A130296 First few rows of the triangle are:
%e A130296 1;
%e A130296 2, 1;
%e A130296 3, 1, 1;
%e A130296 4, 1, 1, 1;
%e A130296 5, 1, 1, 1, 1;
%e A130296 ...
%Y A130296 Cf. A051340, A130297.
%Y A130296 Adjacent sequences: A130293 A130294 A130295 this_sequence A130297 A130298 A130299
%Y A130296 Sequence in context: A065371 A006346 A088742 this_sequence A126705 A113924 A084296
%K A130296 nonn,tabl
%O A130296 1,2
%A A130296 Gary W. Adamson (qntmpkt(AT)yahoo.com), May 20 2007

%I A126705
%S A126705 1,2,1,3,1,1,4,1,1,1,6,1,0,1,1,6,2,1,0,1,1,8,2,1,0,0,1,1,10,2,0,1,0,0,1,
%T A126705 1,10,2,2,1,0,0,0,1,1,10,4,2,0,1,0,0,0,1,1
%N A126705 A097806 * A054523 as infinite lower triangular matrices.
%C A126705 Row sums = (1, 3, 5, 7, 9,...). A129479 = A054523 * A097806. A097806 = the pairwise operator.
%e A126705 First few rows of the triangle are:
%e A126705 1;
%e A126705 2, 1;
%e A126705 3, 1, 1;
%e A126705 4, 1, 1, 1;
%e A126705 6, 1, 0, 1, 1;
%e A126705 6, 2, 1, 0, 1, 1;
%e A126705 8, 2, 1, 0, 0, 1, 1;
%e A126705 ...
%Y A126705 Cf. A097806 * A054523, A129479.
%Y A126705 Adjacent sequences: A126702 A126703 A126704 this_sequence A126706 A126707 A126708
%Y A126705 Sequence in context: A006346 A088742 A130296 this_sequence A113924 A084296 A062534
%K A126705 nonn,tabl
%O A126705 1,2
%A A126705 Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 17 2007

%I A113924
%S A113924 1,1,1,2,1,3,1,1,4,1,1,5,1,2,1,1,1,1,8,1,3,1,1,2,1,1,1,11,2,1,1,1,1,2,1,
%T A113924 1,5,1,4,1,1,1,1,2,19,1,1,1,10,1,7,3,1,2,1,1,1,1,6,1,1,1,5,6,1,1,1,1,2,
%U A113924 1,3,7,1,4,1,1,1,5,2,1
%N A113924 a(n) = GCD(A113605(n+1),A113605(n)). Also, for n >= 2, a(n) = A113605(n+2) - A113605(n-1).
%p A113924 A113605 := proc(n) option remember ; if n <=3 then 1 ; else A113605(n-3)+gcd(A113605(n-1),A113605(n-2)) ; fi ; end: A113924 := proc(n) gcd(A113605(n+1),A113605(n)) ; end; seq(A113924(n),n=1..80) ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 31 2008
%Y A113924 Cf. A113605.
%Y A113924 Adjacent sequences: A113921 A113922 A113923 this_sequence A113925 A113926 A113927
%Y A113924 Sequence in context: A088742 A130296 A126705 this_sequence A084296 A062534 A088425
%K A113924 more,nonn
%O A113924 1,4
%A A113924 Leroy Quet (qq-quet(AT)mindspring.com), Jan 30 2006
%E A113924 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 31 2008

%I A084296
%S A084296 1,2,1,3,1,1,4,1,2,2,5,1,2,2,3,6,2,2,3,2,2,7,3,2,3,3,2,4,8,2,3,2,4,2,3,
%T A084296 2,9,2,3,3,3,2,4,3,4,10,3,3,2,2,2,4,3,3,2,11,1,4,3,2,4,5,4,3,3,4,12,3,3,
%U A084296 4,2,3,6,2,3,5,4,3,13,3,4,2,3,3,3,3,3,3,6,2,4,14,2,3,2,4,5,4,5,3,3,6,4
%N A084296 Triangle: number of distinct prime factors in n-th primorial numbers when n prime factors first appears and in n-1 subsequent integers after.
%C A084296 Omega-values(=A001221) in the subsequent neighborhood of radius n, for primorial numbers are usually neither all distinct or all equal items as it is required in A068069, A045983 sequences.
%e A084296 n-th row of table consists of n numbers A001221[A02110(n+j)], j=0...n-1:
%e A084296 1,
%e A084296 2,1,
%e A084296 3,1,1,
%e A084296 4,1,2,2,
%e A084296 5,1,2,2,3,
%e A084296 6,2,2,3,2,2,
%e A084296 7,3,2,3,3,2,4,
%e A084296 Rows starts with n at indices which are central polygonal numbers:a[A000124(n)]=n; rows ends at a[A000217(n)] terms, at triangular number indices.
%t A084296 lf[x_] := Length[FactorInteger[x]] q[x_] := Apply[Times, Table[Prime[w], {w, 1, x}]] Flatten[Table[Table[lf[q[n]+j], {j, 0, n-1}], {n, 1, 20}], 1]
%Y A084296 Cf. A001221, A002110, A068069, A045983, A000217, A000124.
%Y A084296 Adjacent sequences: A084293 A084294 A084295 this_sequence A084297 A084298 A084299
%Y A084296 Sequence in context: A130296 A126705 A113924 this_sequence A062534 A088425 A010766
%K A084296 nonn,tabl
%O A084296 1,2
%A A084296 Labos E. (labos(AT)ana.sote.hu), May 27 2003

%I A062534
%S A062534 1,2,1,3,1,1,4,2,0,1,5,2,2,1,1,6,3,0,3,2,1,7,3,3,3,5,3,1,8,4,0,6,8,8,4,1,9,4,
%T A062534 4,6,14,16,12,5,1,10,5,0,10,20,30,28,17,6,1,11,5,5,10,30,50,58,45,23,7,1,12,6,0,15,
%U A062534 40,80,108,103,68,30,8,1,13,6,6,15,55,120,188,211,171,98,38,9,1,14,7,0,21,70,175
%V A062534 1,-2,1,3,-1,1,-4,2,0,1,5,-2,2,1,1,-6,3,0,3,2,1,7,-3,3,3,5,3,1,-8,4,0,6,8,8,4,1,9,-4,
%W A062534 4,6,14,16,12,5,1,-10,5,0,10,20,30,28,17,6,1,11,-5,5,10,30,50,58,45,23,7,1,-12,6,0,15,
%X A062534 40,80,108,103,68,30,8,1,13,-6,6,15,55,120,188,211,171,98,38,9,1,-14,7,0,21,70,175
%N A062534 Table by antidiagonals of coefficient of x^k in expansion of 1/((1+x)^2*(1-x)^n).
%F A062534 Each row is partial sum of preceding row, i.e. T(n, k)=T(n-1, k)+T(n, k-1) with T(0, k)=(k+1)*(-1)^k and T(n, 0)=1.
%Y A062534 Rows are effectively (with minor adjustments): A038608, A001057, A027656, A008805, A006918, A002624, A028346. Cf. A058394 which (adjusting for signs and an overlap of three rows) is effectively the continuation of this table for negative n.
%Y A062534 Adjacent sequences: A062531 A062532 A062533 this_sequence A062535 A062536 A062537
%Y A062534 Sequence in context: A126705 A113924 A084296 this_sequence A088425 A010766 A135841
%K A062534 sign,tabl
%O A062534 0,2
%A A062534 Henry Bottomley (se16(AT)btinternet.com), Jun 25 2001

%I A088425
%S A088425 2,1,3,1,1,4,2,1,1,2,1,2,2,1,2,1,1,3,1,1,3,2,1,1,2,1,2,2,1,1,2,1,3,1,1,
%T A088425 2,1,1,1,2,1,2,2,1,3,2,1,2,1,1,1,1,1,1,2,1,2,1,1,3,2,1,1,1,1,3,2,1,1,2,
%U A088425 1,1,2,1,5,1,1,2,1,1,2,2,1,1,1,1,2,2,1,2,2,1,1,1,1,1,2,1,1,1
%N A088425 Number of primes in arithmetic progression starting with 17 and with d=2n.
%C A088425 Arithmetic progression is stopped when next term is not prime. E.g. for n=6 (d=12), a=4, that is 17,29,41,53 are prime, while next term, 65, is not prime.
%t A088425 bb={}; Do[s=1; Do[If[PrimeQ[17+k*d], s=s+1, bb={bb, s}; Break[]], {k, 10}], {d, 2, 200, 2}]; Flatten[bb]
%Y A088425 Cf. A088420, A088421, A088422, A088423, A088424, A088426, A088427, A088428, A088429.
%Y A088425 Adjacent sequences: A088422 A088423 A088424 this_sequence A088426 A088427 A088428
%Y A088425 Sequence in context: A113924 A084296 A062534 this_sequence A010766 A135841 A089178
%K A088425 easy,nonn
%O A088425 1,1
%A A088425 Zak Seidov (zakseidov(AT)yahoo.com), Sep 29 2003

%I A010766
%S A010766 1,2,1,3,1,1,4,2,1,1,5,2,1,1,1,6,3,2,1,1,1,7,3,2,1,1,1,1,8,4,2,2,1,
%T A010766 1,1,1,9,4,3,2,1,1,1,1,1,10,5,3,2,2,1,1,1,1,1,11,5,3,2,2,1,1,1,1,1,1,
%U A010766 12,6,4,3,2,2,1,1,1,1,1,1,13,6,4,3,2,2,1,1,1,1,1,1,1
%N A010766 Triangle of numbers [ n/k ], k=1..n.
%C A010766 Number of times k occurs as divisor of numbers not greater than n. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 19 2004
%C A010766 Viewed as a partition, row n is the smallest partition that contains every partition of n. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Mar 11 2006
%C A010766 Row sums = A006218 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 30 2007
%H A010766 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b010766.txt">Rows n=1..50 of triangle, flattened</a>
%F A010766 G.f.: 1/(1-x)*Sum_(k>=1} x^k/(1-y*x^k). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 05 2004
%F A010766 Triangle A010766 = A000012 * A051731 as infinite lower triangular matrices. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 30 2007
%F A010766 Equals A000012 * A051731 as infinite lower triangular matrices. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 14 2007
%e A010766 1; 2,1; 3,1,1; 4,2,1,1; 5,2,1,1,1; ...
%Y A010766 Another version of A003988.
%Y A010766 Cf. A013942. Also ... A033330, ...
%Y A010766 Cf. A006218, A115725.
%Y A010766 T(n,1)=n, T(n,2)=A008619(n-2) for n>1, T(n,3)=A008620(n-3) for n>2, T(n,4)=A008621(n-4) for n>3, T(n,5)=A002266(n) for n>4, T(n,n)=1.
%Y A010766 Cf. A051731, A006218.
%Y A010766 Cf. A051731, A000012.
%Y A010766 Adjacent sequences: A010763 A010764 A010765 this_sequence A010767 A010768 A010769
%Y A010766 Sequence in context: A084296 A062534 A088425 this_sequence A135841 A089178 A116599
%K A010766 nonn,tabl,easy,nice
%O A010766 1,2
%A A010766 njas

%I A135841
%S A135841 1,2,1,3,1,1,4,2,1,1,5,2,2,1,1,6,3,2,2,1,1,7,3,3,2,2,1,1,8,4,3,3,2,2,1,
%T A135841 1,9,4,4,3,3,2,2,1,1,10,5,4,4,3,3,2,2,1,1
%N A135841 A000012 * A135839 as infinite lower triangular matrices.
%C A135841 Row sums = A024206: (1, 3, 5, 8, 11, 15, 19,...).
%e A135841 First few rows of the triangle are:
%e A135841 1;
%e A135841 2, 1;
%e A135841 3, 1, 1;
%e A135841 4, 2, 1, 1;
%e A135841 5, 2, 2, 1, 1;
%e A135841 6, 3, 2, 2, 1, 1;
%e A135841 7, 3, 3, 2, 2, 1, 1;
%e A135841 ...
%Y A135841 Cf. A135839, A024206.
%Y A135841 Adjacent sequences: A135838 A135839 A135840 this_sequence A135842 A135843 A135844
%Y A135841 Sequence in context: A062534 A088425 A010766 this_sequence A089178 A116599 A138121
%K A135841 nonn,tabl
%O A135841 1,2
%A A135841 Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 01 2007

%I A089178
%S A089178 1,1,1,1,2,1,3,1,1,4,2,1,5,4,1,6,6,1,7,9,1,1,8,12,2,1,9,16,4,1,10,20,6,
%T A089178 1,11,25,10,1,12,30,14,1,13,36,20,1,14,42,26,1,15,49,35,1,1,16,56,44,2,
%U A089178 1,17,64,56,4,1,18,72,68,6,1,19,81,84,10,1,20,90,100,14,1,21,100,120,20
%N A089178 Triangle T(n,k) (n >= 0, 0 <= k <= 1+log_2(floor(n+1)) read by rows: row 0 = {1}, row 1 = {1 1}; for n >=2, row n = row n-1 + (row floor((n-1)/2) shifted one place right).
%H A089178 N. J. A. Sloane and J. A. Sellers, <a href="http://arXiv.org/abs/math.CO/0312418">On non-squashing partitions</a>, Discrete Math., 294 (2005), 259-274.
%F A089178 G.f.: (1/(1-x))*(1+Sum(y^(k+1)*x^(2^(k+1)-1)/Product(1-x^(2^j), j=0..k), k=0..infinity)).
%e A089178 Triangle begins:
%e A089178 1
%e A089178 1 1
%e A089178 1 2
%e A089178 1 3 1
%e A089178 1 4 2
%e A089178 1 5 4
%e A089178 1 6 6
%e A089178 1 7 9 1
%Y A089178 Also obtained by dividing rows of A089177 by "1 1".
%Y A089178 Row sums give A033485.
%Y A089178 Adjacent sequences: A089175 A089176 A089177 this_sequence A089179 A089180 A089181
%Y A089178 Sequence in context: A088425 A010766 A135841 this_sequence A116599 A138121 A138151
%K A089178 nonn,tabf,easy
%O A089178 0,5
%A A089178 njas, Dec 08 2003
%E A089178 More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 10 2003

%I A116599
%S A116599 1,1,1,1,2,1,3,1,1,4,2,1,6,3,1,1,8,4,2,1,11,6,3,1,1,15,8,4,2,1,20,11,6,
%T A116599 3,1,1,26,15,8,4,2,1,35,20,11,6,3,1,1,45,26,15,8,4,2,1,58,35,20,11,6,3,
%U A116599 1,1,75,45,26,15,8,4,2,1,96,58,35,20,11,6,3,1,1,121,75,45,26,15,8,4,2,1
%N A116599 Triangle read by rows: T(n,k) is the number of partitions of n having exactly k parts equal to 2 (n>=0, 0<=k<=floor(n/2)).
%C A116599 Row n has 1+floor(n/2) terms. Row sums are the partition numbers (A000041). T(n,0)=A027336(n), Sum(k*T(n,k),k=0..floor(n/2))=A024786(n). Column k has g.f. x^(2k)/[(1-x)product(1-x^j,j=3..infinity)] (k=0,1,2,...).
%F A116599 G.f.=1/[(1-x)(1-tx^2)product(1-x^j, j=3..infinity)]. T(n,k)=p(n-2k)-p(n-2k-2) for k<=(n-2)/2; T(n, floor(n/2))=1 (follows at once from the g.f.).
%e A116599 T(6,1)=3 because we have [4,2], [3,2,1], and [2,1,1,1,1].
%e A116599 Triangle starts:
%e A116599 1;
%e A116599 1;
%e A116599 1,1;
%e A116599 2,1;
%e A116599 3,1,1;
%e A116599 4,2,1;
%e A116599 6,3,1,1;
%e A116599 8,4,2,1;
%p A116599 with(combinat): T:=proc(n,k) if k=floor(n/2) then 1 elif k<=(n-2)/2 then numbpart(n-2*k)-numbpart(n-2*k-2) fi end: for n from 0 to 18 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%Y A116599 Cf. A000041, A027336, A024786.
%Y A116599 Adjacent sequences: A116596 A116597 A116598 this_sequence A116600 A116601 A116602
%Y A116599 Sequence in context: A010766 A135841 A089178 this_sequence A138121 A138151 A122610
%K A116599 nonn,tabl
%O A116599 0,5
%A A116599 Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 18 2006

%I A138121
%S A138121 1,2,1,3,1,1,4,2,2,1,1,1,5,3,2,1,1,1,1,1,6,3,3,4,2,2,2,2,1,1,1,1,1,1,1,
%T A138121 7,4,3,5,2,3,2,2,1,1,1,1,1,1,1,1,1,1,1,8,4,4,5,3,6,2,3,3,2,4,2,2,2,2,2,
%U A138121 2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,9,5,4,6,3,3,3,3,7,2,4,3,2,5,2,2,3,2,2
%N A138121 A shell model of partitions. Triangle read by rows: row n lists the parts of the outer shell of the partitions of n.
%C A138121 Row n lists the terms of row n of A135010, in decreasing order. See the integrated diagram of partitions and more information in the entry A135010.
%e A138121 Triangle begins:
%e A138121 1
%e A138121 2,1
%e A138121 3,1,1
%e A138121 4,2,2,1,1,1
%e A138121 5,3,2,1,1,1,1,1,
%e A138121 6,3,3,4,2,2,2,2,1,1,1,1,1,1,1
%e A138121 7,4,3,5,2,3,2,2,1,1,1,1,1,1,1,1,1,1,1
%e A138121 8,4,4,5,3,6,2,3,3,2,4,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%e A138121 9,5,4,6,3,3,3,3,7,2,4,3,2,5,2,2,3,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%Y A138121 Cf. A135010, A138122, A138123, A138136, A138137, A138138, A138151, A138152, A138153.
%Y A138121 Adjacent sequences: A138118 A138119 A138120 this_sequence A138122 A138123 A138124
%Y A138121 Sequence in context: A135841 A089178 A116599 this_sequence A138151 A122610 A011973
%K A138121 nonn
%O A138121 1,2
%A A138121 Omar E. Pol (info(AT)polprimos.com), Mar 21 2008

%I A138151
%S A138151 1,2,1,3,1,1,4,2,2,1,1,1,5,3,2,1,1,1,1,1,6,4,2,3,3,2,2,2,1,1,1,1,1,1,1,
%T A138151 7,5,2,4,3,3,2,2,1,1,1,1,1,1,1,1,1,1,1,8,6,2,5,3,4,4,4,2,2,3,3,2,2,2,2,
%U A138151 2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,9,7,2,6,3,5,4,5,2,2,4,3,2,3,3,3,3,2,2
%N A138151 A shell model of partitions. Triangle read by rows: row n lists the parts of the outer shell of the partitions of n.
%C A138151 Row n lists the terms of row n of A138138, in decreasing order. See the integrated diagram of partitions and more information in the entry A138138.
%e A138151 Triangle begins:
%e A138151 1
%e A138151 2,1
%e A138151 3,1,1
%e A138151 4,2,2,1,1,1
%e A138151 5,3,2,1,1,1,1,1,
%e A138151 6,4,2,3,3,2,2,2,1,1,1,1,1,1,1
%e A138151 7,5,2,4,3,3,2,2,1,1,1,1,1,1,1,1,1,1,1
%e A138151 8,6,2,5,3,4,4,4,2,2,3,3,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%e A138151 9,7,2,6,3,5,4,5,2,2,4,3,2,3,3,3,3,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%Y A138151 Cf. A135010, A138121, A138122, A138123, A138136, A138137, A138138, A138152, A138153.
%Y A138151 Adjacent sequences: A138148 A138149 A138150 this_sequence A138152 A138153 A138154
%Y A138151 Sequence in context: A089178 A116599 A138121 this_sequence A122610 A011973 A115139
%K A138151 nonn
%O A138151 1,2
%A A138151 Omar E. Pol (info(AT)polprimos.com), Mar 21 2008

%I A122610
%S A122610 1,1,2,1,3,1,1,4,3,1,1,5,6,1,2,1,6,10,1,6,1,1,7,15,6,11,6,1,1,8,21,15,
%T A122610 15,18,1,2,1,9,28,29,15,39,6,9,1,1,10,36,49,7,69,30,21,9,1,1,11,45,76,
%U A122610 14,105,84,30,36,1,2,1,12,55,111,54,140,182,15,96,14,12,1,1,13,66,155
%V A122610 1,1,-2,1,-3,1,1,-4,3,1,1,-5,6,1,-2,1,-6,10,-1,-6,1,1,-7,15,-6,-11,6,1,1,-8,21,-15,-15,
%W A122610 18,1,-2,1,-9,28,-29,-15,39,-6,-9,1,1,-10,36,-49,-7,69,-30,-21,9,1,1,-11,45,-76,14,105,
%X A122610 -84,-30,36,1,-2,1,-12,55,-111,54,140,-182,-15,96,-14,-12,1,1,-13,66,-155
%N A122610 Triangle read by rows: T(n,k) is coefficient of x^k in Sum_{m=0..n} x^m*(1-x)^(n-m)*(-1)^[(m+1)/2]*binomial(m-[(m+1)/2],[m/2]).
%D A122610 P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
%e A122610 1
%e A122610 1, -2
%e A122610 1,-3, 1
%e A122610 1, -4, 3, 1
%e A122610 1, -5, 6, 1,-2
%e A122610 1, -6, 10,-1, -6, 1
%e A122610 1, -7, 15, -6, -11, 6, 1
%e A122610 1,-8, 21, -15, -15, 18, 1, -2
%t A122610 T[n_, k_] := (-1)^Floor[(k + 1)/2]*Binomial[n - Floor[(k + 1)/2], Floor[k/2]] a = Table[CoefficientList[Sum[T[n, k]*p^k*(1 - p)^(n -k), {k, 0, n}], p], {n, 0, 10}]; Flatten[a]
%o A122610 (PARI) {T(n,k)=local(A); if(k<0|k>n, 0, A=sum(k=0, n, x^k*(1-x)^(n-k)*(-1)^((k+1)\2)*binomial(n-((k+1)\2),k\2)); polcoeff(A,k))}
%Y A122610 Cf. A066170.
%Y A122610 Adjacent sequences: A122607 A122608 A122609 this_sequence A122611 A122612 A122613
%Y A122610 Sequence in context: A116599 A138121 A138151 this_sequence A011973 A115139 A124033
%K A122610 sign,tabl
%O A122610 1,3
%A A122610 Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Sep 20 2006
%E A122610 Edited by njas, Sep 24 2006

%I A011973
%S A011973 1,1,1,1,1,2,1,3,1,1,4,3,1,5,6,1,1,6,10,4,1,7,15,10,1,1,8,21,20,5,1,9,
%T A011973 28,35,15,1,1,10,36,56,35,6,1,11,45,84,70,21,1,1,12,55,120,126,56,7,1,
%U A011973 13,66,165,210,126,28,1,1,14,78,220,330,252,84,8,1,15,91,286,495,462
%N A011973 Triangle of numbers {C(n-k,k), n >= 0, 0<=k<=[ n/2 ]}; or, triangle of coefficients of (one version of) Fibonacci polynomials.
%C A011973 T(n,k) is the number of subsets of {1,2,...,n-1} of size k and containing no consecutive integers. Example: T(6,2)=6 because the subsets of size 2 of {1,2,3,4,5} with no consecutive integers are {1,3},{1,4},{1,5},{2,4},{2,5}, and {3,5}. Equivalently, T(n,k) is the number of k-matchings of the path graph P_n. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 10 2003
%C A011973 T(n,k)= number of compositions of n+2 into k+1 parts, all >=2. Example: T(6,2)=6 because we have (2,2,4),(2,4,2),(4,2,2),(2,3,3),(3,2,3) and (3,3,2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2005
%C A011973 Given any recurrence sequence S(k) = x*a(k-1) + a(k-2), starting (1, x, x^2+1,...); the (k+1)-th term of the series = f(x) in the k-th degree polynomial: (1, (x), (x^2 + 1), (x^3 + 2x), (x^4 + 3x^2 + 1), (x^5 + 4x^3 + 3x), (x^6 + 5x^4 + 6x^2 + 1),...Example: let x = 2, then S(k) = 1, 2, 5, 12, 29, 70, 169,...such that A000129(7) = 169 = f(x), x^6 + 5x^4 + 6x^2 + 1 = (64 + 80 + 24 + 1). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 16 2008
%D A011973 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 141ff.
%D A011973 A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 91, 145.
%D A011973 C. D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, 1993.
%D A011973 A. Holme, A combinatorial proof ..., Discrete Math., 241 (2001), 363-378; see p. 375.
%D A011973 C.-K. Lim and K. S. Lam, The characteristic polynomial of ladder graphs and an annihilating uniqueness theorem, Discr. Math., 151 (1996), 161-167.
%D A011973 D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, Vol. A 99 (2002), 307-344 (Table 3).
%H A011973 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b011973.txt">Rows n=0..100 of triangle, flattened</a>
%H A011973 H. S. Wilf, <a href="http://www.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a>, 2nd edn., Academic Press, NY, 1994, p. 26, ex. 12.
%H A011973 <a href="http://www.research.att.com/~njas/sequences/Sindx_Pas.html#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A011973 Let F(n, x) be the n-th Fibonacci polynomial in x; the g.f. for F(n, x) is sum_{n=0..inf} F(n, x)*y^n = (1 + x*y)/(1 - y - x*y^2). - Paul D. Hanna (pauldhanna(AT)juno.com)
%F A011973 T(m, n) = 0 for n /= 0 and m <= 1 T(0, 0) = T(1, 0) = 1 T(m, n) = T(m - 1, n) + T(m-2, n-1) for m >= 2 (i.e. like the recurrence for Pascal's triangle A007318, but going up one row as well as left one column for the second summand). E.g. T(7, 2) = 10 = T(6, 2) + T(5, 1) = 6 + 4 - Rob Arthan (rda(AT)lemma-one.com), Sep 22 2003
%F A011973 G.f. for k-th column: x^(2k-1)/(1-x)^(k+1).
%F A011973 Identities for the Fibonacci polynomials F(n, x):
%F A011973 F(m+n+1, x) = F(m+1, x)*F(n+1, x) + x*F(m, x)F(n, x).
%F A011973 F(n, x)^2-F(n-1, x)*F(n+1, x)=(-x)^(n-1).
%F A011973 The degree of F(n, x) is floor((n-1)/2) and F(2p, x) = F(p, x) times a polynomial of equal degree which is 1 mod p.
%e A011973 {1}; {1}; {1,1}; {1,2}; {1,3,1}; {1,4,3}; {1,5,6,1}; {1,6,10,4}, ...
%e A011973 The first few Fibonacci polynomials (defined here by F(0,x) = 0, F(1,x) = 1; F(n+1, x) = F(n, x) + x*F(n-1, x)) are:
%e A011973 0: 0
%e A011973 1: 1
%e A011973 2: 1
%e A011973 3: 1 + x
%e A011973 4: 1 + 2 x
%e A011973 5: 1 + 3 x + x^2
%e A011973 6: (1 + x) (1 + 3 x)
%e A011973 7: 1 + 5 x + 6 x^2 + x^3
%e A011973 8: (1 + 2 x) (1 + 4 x + 2 x^2 )
%e A011973 9: (1 + x) (1 + 6 x + 9 x^2 + x^3 )
%e A011973 10: (1 + 3 x + x^2 ) (1 + 5 x + 5 x^2 )
%e A011973 11: 1 + 9 x + 28 x^2 + 35 x^3 + 15 x^4 + x^5
%p A011973 a := proc(n) local k; [ seq(binomial(n-k,k),k=0..floor(n/2)) ]; end;
%o A011973 (PARI) T(n,k)=if(k<0|k>n,0,binomial(n-k,k))
%Y A011973 Row sums = A000045(n+1) (Fibonacci numbers).
%Y A011973 Cf. A054123.
%Y A011973 Adjacent sequences: A011970 A011971 A011972 this_sequence A011974 A011975 A011976
%Y A011973 Sequence in context: A138121 A138151 A122610 this_sequence A115139 A124033 A112543
%Y A011973 Cf. A000129.
%K A011973 tabf,easy,nonn,nice,new
%O A011973 0,6
%A A011973 njas

%I A115139
%S A115139 1,1,1,1,1,2,1,3,1,1,4,3,1,5,6,1,1,6,10,4,1,7,15,10,1,1,8,21,20,5,1,9,28,
%T A115139 35,15,1,1,10,36,56,35,6,1,11,45,84,70,21,1,1,12,55,120,126,56,7,1,13,66,
%U A115139 165,210,126,28,1,1,14,78,220,330,252,84,8,1,15,91,286,495,462,210,36,1
%V A115139 1,1,1,-1,1,-2,1,-3,1,1,-4,3,1,-5,6,-1,1,-6,10,-4,1,-7,15,-10,1,1,-8,21,-20,5,1,-9,28,
%W A115139 -35,15,-1,1,-10,36,-56,35,-6,1,-11,45,-84,70,-21,1,1,-12,55,-120,126,-56,7,1,-13,66,
%X A115139 -165,210,-126,28,-1,1,-14,78,-220,330,-252,84,-8,1,-15,91,-286,495,-462,210,-36,1
%N A115139 Array of coefficients of polynomials related to integer powers of the generating function of Catalan numbers A000108.
%C A115139 This is a signed version of A011973 (Fibonacci polynomials) with different offset.
%C A115139 The sequence of row lengths is [1,1,2,2,3,3,4,4,5,5,6,6,...]=A008619(n-1), n>=1.
%C A115139 The row sums give the period 6 sequence [1,1,0,-1,-1,0,...]=A010892(n-1), n>=1.
%C A115139 The o.g.f. for the column m sequence (with leading zeros) is ((-1)^m)*x^(2*m+1)/(1-x)^(m+1).
%C A115139 The unsigned row sums give the Fibonacci numbers A000045(n-1). n>=1.
%C A115139 The row polynomial are P(n,x):= sum(a(n,m)*x^m,m=0..ceil(n/2)-1) = (sqrt(x)^(n-1))*Sn(n-1,1/sqrt(x))), n>=1, with Chebyshev's S(n,x) polynomials A049310.
%C A115139 These polynomials appear in the formula 1/c(x)^n = P(n+1,x) - x*P(n,x)*c(x), n>=1, with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers). See the W. Lang reference, eqs. (1) and (2), p. 408, with P(n,x):=p(-n,x).
%C A115139 These polynomials appear also in the formula c(x)^n = (-P(n-1,x) + P(n,x)*c(x))/x^(n-1), n>=1, with the above given o.g.f. c(x) of A000108 (Catalan numbers). See the W. Lang reference, eq. (1), with P(n,x):=p(-n,x).
%C A115139 With offset n>=0 this array a(n,m) coincides with the row reversed coefficient table of Chebyshev's S-polynomials without interspersed zeros. See A049310 for the S(n,x) coefficient table with increasing powers of x.
%D A115139 W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419.
%H A115139 W. Lang: <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A115139.text">First 16 rows.</a>
%F A115139 a(n, m)= ((-1)^(m))*binomial(n-1-m, m), n>=1, m=0..ceil(n/2)-1.
%F A115139 a(n,m)=[x^m]P(n,x), n>=1, m=0..ceil(n/2)-1, with P(n,x) given above in terms of Chebyshev's S-polynomials.
%e A115139 [1];[1];[1,-1];[1,-2];[1,-3,1];[1,-4,3];[1,-5,6,-1];...
%e A115139 1/c(x) = P(2,x) - x*P(1,x)*c(x) = 1 - x*c(x), with the o.g.f.
%e A115139 of A000108 (Catalan).
%e A115139 1/c(x)^2 = P(3,x) - x*P(2,x)*c(x) = (1-x) - x*c(x).
%e A115139 c(x)^2 = (-P(1,x) + P(2,x)*c(x))/x^1 = (-1 + 1*c(x))/x.
%e A115139 c(x)^3 = (-P(2,x) + P(3,x)*c(x))/x^2 = (-1 + (1-x)*c(x))/x^2.
%e A115139 P(3,x)=1-x = x*S(2,1/sqrt(x))) with Chebyshev's S(2,y) = U(2,y/2) = y^2-1.
%Y A115139 Adjacent sequences: A115136 A115137 A115138 this_sequence A115140 A115141 A115142
%Y A115139 Sequence in context: A138151 A122610 A011973 this_sequence A124033 A112543 A099478
%K A115139 sign,easy,tabf
%O A115139 1,6
%A A115139 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Jan 13 2006

%I A124033
%S A124033 1,1,1,1,2,1,3,1,1,4,3,1,5,6,1,1,6,10,4,1,7,15,10,1,1,8,21,20,5,1,9,28,
%T A124033 35,15,1,1,10,36,56,35,6,1,11,45,84,70,21,1,1,12,55,120,126,56,7,1,13,66,
%U A124033 165,210,126,28,1,1,14,78,220,330,252,84,8,1,15,91,286,495,462,210,36,1
%V A124033 1,1,-1,1,-2,1,-3,1,1,-4,3,1,-5,6,-1,1,-6,10,-4,1,-7,15,-10,1,1,-8,21,-20,5,1,-9,28,
%W A124033 -35,15,-1,1,-10,36,-56,35,-6,1,-11,45,-84,70,-21,1,1,-12,55,-120,126,-56,7,1,-13,66,
%X A124033 -165,210,-126,28,-1,1,-14,78,-220,330,-252,84,-8,1,-15,91,-286,495,-462,210,-36,1
%N A124033 Triangle read by rows: T(n,k) is the coefficient of x^k in the determinant of the n X n tridiagonal matrix with 1's on the main diagonal and x^(1/2) on the sub- and superdiagonal (n>=1, 0<=k<=floor(n/2)).
%C A124033 Row n contains 1+floor(n/2) terms. With x on the main diagonal and 1's on the sub- and superdiagonal, one obtains A049310.
%e A124033 Setting y=sqrt(x), we have det(matrix({1,y,0,0},{y,1,y,0},{0,y,1,y},{0,0,y,1}))=1-3x+x^2, leading to row 4 of the triangle.
%e A124033 Triangle starts:
%e A124033 1;
%e A124033 1,-1;
%e A124033 1,-2;
%e A124033 1,-3,1;
%e A124033 1,-4,3;
%e A124033 1,-5,6,-1;
%e A124033 1,-6,10,-4;
%p A124033 with(linalg): m:=proc(i,j) if i=j then 1 elif abs(i-j)=1 then sqrt(x) else 0 fi end: T:=(n,k)->coeff(det(matrix(n,n,m)),x,k): for n from 1 to 16 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form
%Y A124033 Cf. A049310.
%Y A124033 Adjacent sequences: A124030 A124031 A124032 this_sequence A124034 A124035 A124036
%Y A124033 Sequence in context: A122610 A011973 A115139 this_sequence A112543 A099478 A133913
%K A124033 sign,tabf
%O A124033 1,5
%A A124033 Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Nov 01 2006
%E A124033 Edited by njas, Dec 03 2006

%I A112543
%S A112543 1,2,1,3,1,1,4,3,2,1,5,2,1,1,1,6,5,4,3,2,1,7,3,5,1,3,1,1,8,7,2,5,4,1,2,
%T A112543 1,9,4,7,3,1,2,3,1,1,10,9,8,7,6,5,4,3,2,1,11,5,3,2,7,1,5,1,1,1,1,12,11,
%U A112543 10,9,8,7,6,5,4,3,2,1,13,6,11,5,9,4,1,3,5,2,3,1,1,14,13,4,11,2,3,8,7,2
%N A112543 Numerators of fractions n/m in array by anti-diagonals.
%H A112543 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ExponentialIntegral.html">Exponential Integral</a>
%F A112543 For n/m: g.f.: x/(1-x)*log(1/(1-y)), e.g.f.: x*e^x*(Ei(y)-log(y)+gamma) = x*e^x*integral_t=0^y (e^t-1)dt.
%e A112543 a(2,4)=1/2 because 2/4 = 1/2.
%o A112543 (PARI) t1(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1 t2(n)=n-binomial(floor(1/2+sqrt(2*n)),2) vector(100,n,t1(n)/gcd(t1(n),t2(n)))
%Y A112543 Denominators in A112544. Reduced version of A004736/A002260.
%Y A112543 Adjacent sequences: A112540 A112541 A112542 this_sequence A112544 A112545 A112546
%Y A112543 Sequence in context: A011973 A115139 A124033 this_sequence A099478 A133913 A026807
%K A112543 easy,frac,nonn
%O A112543 1,2
%A A112543 Frank Adams-Watters (FrankTAW(AT)Netscape.net), Sep 11 2005

%I A099478
%S A099478 2,1,3,1,1,4,3,6,1,1,4,2,9,4,9,14,4,1,3,4,36,5,25,4,10,4,18,3,21,9,9,21,
%T A099478 16,65,12,8,51,1,22,2,30,6,10,63,1,30,15,3,10,1,22,57,202,4,3,53,1,34,
%U A099478 12,10,22,29,28,31,7,6,70,29,16,94,37,51,30,56,19,23,70,50,99,16,34,5
%N A099478 Least a(n) such that a(n)*2^n*(2^n-1)-1 is prime.
%e A099478 1*2^6*(2^6-1)-1=4031=29*139
%e A099478 2*2^6*(2^6-1)-1=8063=11*733
%e A099478 3*2^6*(2^6-1)-1=12095=5*2419
%e A099478 4*2^6*(2^6-1)-1=16127 prime so a(6)=4
%Y A099478 Adjacent sequences: A099475 A099476 A099477 this_sequence A099479 A099480 A099481
%Y A099478 Sequence in context: A115139 A124033 A112543 this_sequence A133913 A026807 A106740
%K A099478 easy,nonn
%O A099478 1,1
%A A099478 Pierre CAMI (pierrecami(AT)tele2.fr), Nov 18 2004

%I A133913
%S A133913 1,1,2,1,3,1,1,4,4,2,1,5,8,6,3,1,6,13,14,9,1,1,7,19,27,23,10,2,1,8,26,
%T A133913 46,50,33,12,1,1,9,34,72,96,83,45,13,2,1,10,43,106,168,179,128,58,15,3,
%U A133913 1,11,53,149,274,347,307,186,73,18,1,1,12,64,202,423,621,654,493,259,91
%N A133913 Triangle, antidiagonals of an array generated from partial sums of A007001.
%C A133913 Row sums = A133914: (1, 3, 5, 11, 23, 44, 89, 177, 355,...). Right border = A007001: (1, 2, 1, 2, 3, 1, 2, 1,...).
%F A133913 Given A007001: (1, 2, 1, 2, 3, 1, 2, 1,...) as first row of an array, n-th row = partial sum sequence of (n-1)-th row. The Triangle A133913 = the antidiagonals of this array.
%e A133913 First few rows of the array are:
%e A133913 1, 2, 1, 2, 3, 1, 2,...
%e A133913 1, 3, 4, 6, 9, 10, 12,...
%e A133913 1, 4, 8, 14, 23, 33, 45,...
%e A133913 1, 5, 13, 27, 50, 83, 128,...
%e A133913 1, 6, 19, 46, 96, 179, 307,...
%e A133913 ...
%e A133913 First few rows of the triangle are:
%e A133913 1;
%e A133913 1, 2;
%e A133913 1, 3, 1;
%e A133913 1, 4, 4, 2;
%e A133913 1, 5, 8, 6, 3;
%e A133913 1, 6, 13, 14, 9, 1;
%e A133913 1, 7, 19, 27, 23, 10, 2;
%e A133913 1, 8, 26, 46, 50, 33, 12, 1;
%e A133913 1, 9, 34, 72, 96, 83, 45, 13, 2;
%e A133913 1, 10, 43, 106, 168, 179, 128, 58, 15, 3;
%e A133913 ...
%Y A133913 Cf. A007001, A133912, A133914.
%Y A133913 Adjacent sequences: A133910 A133911 A133912 this_sequence A133914 A133915 A133916
%Y A133913 Sequence in context: A124033 A112543 A099478 this_sequence A026807 A106740 A110619
%K A133913 nonn,tabl
%O A133913 1,3
%A A133913 Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 28 2007

%I A026807
%S A026807 1,2,1,3,1,1,5,2,1,1,7,2,1,1,1,11,4,2,1,1,1,15,4,2,1,1,1,1,22,7,3,2,1,1,
%T A026807 1,1,30,8,4,2,1,1,1,1,1,42,12,5,3,2,1,1,1,1,1,56,14,6,3,2,1,1,1,1,1,1,
%U A026807 77,21,9,5,3,2,1,1,1,1,1,1,101,24,10,5,3,2,1,1,1,1,1,1,1,135,34,13
%N A026807 Triangular array T read by rows: T(n,k) = number of partitions of n in which every part is >=k, for k=1,2,...,n.
%C A026807 T(n,1)=A000041(n), T(n,2)=A002865(n) for n>1, T(n,3)=A008483(n) for n>2, T(n,4)=A008484(n) for n>3.
%F A026807 G.f.: Sum_{k>=1} y^k*(-1+1/Product_{i>=0} (1-x^(k+i))). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jun 22 2003
%F A026807 T(n, k) = T(n, k+1)+T(n-k, k) (where T(n, n) = 1). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jan 24 2005
%F A026807 Equals A026794 * A000012 as infinite lower triangular matrices. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 31 2008
%e A026807 Sum_{k>=1} y^k*(-1+1/Product_{i>=0} (1-x^(k+i))) =
%e A026807 y*x+(2*y+y^2)*x^2+(3*y+y^2+y^3)*x^3+(5*y+2*y^2+y^3+y^4)*x^4+(7*y+2*y^2+y^3+y^4+y^5)*x^5+...
%Y A026807 Row sums give A046746.
%Y A026807 Cf. A026835.
%Y A026807 Cf. A026794.
%Y A026807 Adjacent sequences: A026804 A026805 A026806 this_sequence A026808 A026809 A026810
%Y A026807 Sequence in context: A112543 A099478 A133913 this_sequence A106740 A110619 A129761
%K A026807 nonn,tabl
%O A026807 1,2
%A A026807 Clark Kimberling (ck6(AT)evansville.edu)

%I A106740
%S A106740 2,1,3,1,1,5,2,1,1,8,1,1,1,1,13,1,3,1,1,1,21,2,1,1,2,1,1,34,1,1,5,1,1,1,
%T A106740 1,55,1,1,1,1,1,1,1,1,89,2,3,1,8,1,3,2,1,1,144,1,1,1,1,1,1,1,1,1,1,233,
%U A106740 1,1,1,1,13,1,1,1,1,1,1,377,2,1,5,2,1,1,2,5,1,2,1,1,610,1,3,1,1,1,21,1
%N A106740 Triangle read by rows: greatest common divisors of pairs of Fibonacci numbers greater than 1: T(n,k) = GCD(Fib(n),Fib(k)), 2<k<n.
%C A106740 T(n,k) = GCD(A000045(n), A000045(k)), 2<k<n;
%C A106740 T(n,3) = abs(A061347(n)) for n>2;
%C A106740 T(n,4) = A093148(n) for n>3;
%C A106740 T(n,n) = A000045(n).
%Y A106740 Adjacent sequences: A106737 A106738 A106739 this_sequence A106741 A106742 A106743
%Y A106740 Sequence in context: A099478 A133913 A026807 this_sequence A110619 A129761 A123864
%K A106740 nonn,tabl
%O A106740 1,1
%A A106740 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 15 2005

%I A110619
%S A110619 1,2,1,3,1,1,5,3,1,1,7,3,1,1,1,11,7,4,1,1,1,15,8,4,1,1,1,1,22,15,5,5,1,
%T A110619 1,1,1,30,18,12,5,1,1,1,1,1,42,30,14,6,6,1,1,1,1,1,56,37,16,6,6,1,1,1,1,
%U A110619 1,1,77,58,34,19,7,7,1,1,1,1,1,1,101,71,39,21,7,7,1,1,1,1,1,1,1,135,105
%N A110619 Triangle of number of partitions of n with no part more than n/k; also partitions of n into n/k or fewer parts.
%F A110619 T(n, k)=A008284(n+floor[n/k], floor[n/k]). T(0, k)=1; T(n, k)=0 for 0<n<k; T(n, k)=1 for k<=n<2k; T(n, 1)=A000041(n); T(n, 2)=A110618(n).
%e A110619 Rows start: 1; 2,1; 3,1,1; 5,3,1,1; 7,3,1,1,1; 11,7,4,1,1,1; etc.
%e A110619 T(7,3)=4 since 7 can be partitioned as 1+1+1+1+1+1+1, 2+1+1+1+1+1, 2+2+1+1+1, or 2+2+2+1, and also as 7, 6+1, 5+2, or 4+3.
%Y A110619 First column is A000041, second is A110618.
%Y A110619 Adjacent sequences: A110616 A110617 A110618 this_sequence A110620 A110621 A110622
%Y A110619 Sequence in context: A133913 A026807 A106740 this_sequence A129761 A123864 A035175
%K A110619 nonn,tabl
%O A110619 1,2
%A A110619 Henry Bottomley (se16(AT)btinternet.com), Aug 01 2005

%I A129761
%S A129761 1,2,1,3,1,1,6,1,1,2,1,11,1,1,2,1,3,1,1,22,1,1,2,1,3,1,1,6,1,1,2,1,43,1,
%T A129761 1,2,1,3,1,1,6,1,1,2,1,11,1,1,2,1,3,1,1,86,1,1,2,1,3,1,1,6,1,1,2,1,11,1,
%U A129761 1,2,1,3,1,1,22,1,1,2,1,3,1,1,6,1,1,2,1,171,1,1,2,1,3,1,1,6
%N A129761 Differences between adjacent Fibbinary numbers (A003714).
%C A129761 Conjecture: All members are of the form ceiling(2^k/3) (A005578).
%Y A129761 Adjacent sequences: A129758 A129759 A129760 this_sequence A129762 A129763 A129764
%Y A129761 Sequence in context: A026807 A106740 A110619 this_sequence A123864 A035175 A106406
%K A129761 nonn
%O A129761 1,2
%A A129761 Ralf Stephan, May 14 2007

%I A123864
%S A123864 1,1,2,1,3,1,2,0,4,1,2,0,3,0,0,1,5,2,2,2,3,0,0,2,4,1,0,1,0,0,2,2,6,0,4,
%T A123864 0,3,0,4,0,4,0,0,0,0,1,4,2,5,1,2,2,0,2,2,0,0,2,0,0,3,2,4,0,7,0,0,0,6,2,
%U A123864 0,0,4,0,0,1,6,0,0,2,5,1,0,2,0,2,0,0,0,0,2,0,6,2,4,2,6,0,2,0,3,0,4,0,0
%N A123864 Expansion of (eta(q^3)eta(q^5))^2/(eta(q)eta(q^15)) in powers of q.
%F A123864 Euler transform of period 15 sequence [ 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -2, ...].
%F A123864 Moebius transform is period 15 sequence [ 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1, 0, ...].
%F A123864 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=-v^3+4uvw-2uw^2-u^2w.
%F A123864 G.f.: Product_{k>0} ((1-x^(3k))*(1-x^(5k)))^2/((1-x^k)*(1-x^(15k))).
%F A123864 G.f.: (1/2)(Sum_{n,m} x^(n^2+n*m+4*m^2) +x^(2*n^2+n*m+2*m^2)).
%F A123864 a(15n+7)=a(15n+11)=a(15n+13)=a(15n+14)=0.
%o A123864 (PARI) {a(n)=if(n<1, n==0, sumdiv(n, d, kronecker(-15,d)))}
%o A123864 (PARI) {a(n)=if(n<1, n==0, (qfrep([2, 1;1, 8],n, 1)+qfrep([4, 1;1, 4], n, 1))[n])}
%o A123864 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^3+A)*eta(x^5+A))^2/(eta(x+A)*eta(x^15+A)), n))}
%Y A123864 A035175(n)=a(n) if n>0.
%Y A123864 Adjacent sequences: A123861 A123862 A123863 this_sequence A123865 A123866 A123867
%Y A123864 Sequence in context: A106740 A110619 A129761 this_sequence A035175 A106406 A092412
%K A123864 nonn
%O A123864 0,3
%A A123864 Michael Somos, Oct 14 2006

%I A035175
%S A035175 1,2,1,3,1,2,0,4,1,2,0,3,0,0,1,5,2,2,2,3,0,0,2,4,1,0,1,0,0,2,2,6,0,4,0,
%T A035175 3,0,4,0,4,0,0,0,0,1,4,2,5,1,2,2,0,2,2,0,0,2,0,0,3,2,4,0,7,0,0,0,6,2,0,
%U A035175 0,4,0,0,1,6,0,0,2,5,1,0,2,0,2,0,0,0,0,2,0,6,2,4,2,6,0,2,0,3,0,4,0,0,0
%N A035175 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -15.
%F A035175 Expansion of -1+(eta(q^3)eta(q^5))^2/(eta(q)eta(q^15)) in powers of q. - Michael Somos Aug 25 2006
%F A035175 Euler transform of period 15 sequence [ 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -2, ...]. if a(0)=1. - Michael Somos Aug 25 2006
%F A035175 Moebius transform is period 15 sequence [ 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1, 0, ...]. - Michael Somos Aug 25 2006
%F A035175 Given g.f. A(x), then B(x)=1+A(x) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=-v^3+4uvw-2uw^2-u^2w.
%F A035175 G.f.: -1+x*Product_{k>0} ((1-x^(3k))(1-x^(5k)))^2/((1-x^k)(1-x^(15k))) . - Michael Somos Aug 25 2006
%F A035175 G.f.: -1+(1/2)(Sum_{n,m} x^(n^2+nm+4m^2) +x^(2n^2+nm+2m^2)). - Michael Somos Aug 25 2006
%F A035175 a(n) is multiplicative with a(3^e) = a(5^e) = 1, a(p^e) = (1+(-1)^e)/2 if p == 7, 11, 13, 14 (mod 15), a(p^e) = e+1 if p == 1, 2, 4, 8 (mod 15). - Michael Somos Aug 25 2006
%F A035175 a(15n+7)=a(15n+11)=a(15n+13)=a(15n+14)=0.
%e A035175 q + 2*q^2 + q^3 + 3*q^4 + q^5 + 2*q^6 + 4*q^8 + q^9 + 2*q^10 +...
%o A035175 (PARI) direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
%o A035175 (PARI) {a(n)=if(n<1, 0, sumdiv(n, d, kronecker(-15,d)))} /* Michael Somos Aug 25 2006 */
%o A035175 (PARI) {a(n)=local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==3|p==5, 1, if((p%15)!=2^valuation(p%15,2), (e+1)%2, (e+1))))))} /* Michael Somos Aug 25 2006 */
%o A035175 (PARI) {a(n)=if(n<1, 0, (qfrep([2, 1;1, 8],n, 1)+qfrep([4, 1;1, 4], n, 1))[n])} /* Michael Somos Aug 25 2006 */
%o A035175 (PARI) {a(n)=local(A); if(n<1, 0, A=x*O(x^n); polcoeff( eta(x^3+A)^2*eta(x^5+A)^2/eta(x+A)/eta(x^15+A), n))} /* Michael Somos Aug 25 2006 */
%Y A035175 Adjacent sequences: A035172 A035173 A035174 this_sequence A035176 A035177 A035178
%Y A035175 Sequence in context: A110619 A129761 A123864 this_sequence A106406 A092412 A078734
%K A035175 nonn,mult
%O A035175 1,2
%A A035175 njas

%I A106406
%S A106406 1,2,1,3,1,2,0,4,1,2,0,3,0,0,1,5,2,2,2,3,0,0,2,4,1,0,1,0,0,2,2,6,0,4,0,
%T A106406 3,0,4,0,4,0,0,0,0,1,4,2,5,1,2,2,0,2,2,0,0,2,0,0,3,2,4,0,7,0,0,0,6,2,0,
%U A106406 0,4,0,0,1,6,0,0,2,5,1,0,2,0,2,0,0,0,0,2,0,6,2,4,2,6,0,2,0,3,0,4,0,0,0
%V A106406 1,-2,-1,3,-1,2,0,-4,1,2,0,-3,0,0,1,5,-2,-2,2,-3,0,0,-2,4,1,0,-1,0,0,-2,2,-6,0,4,0,3,0,
%W A106406 -4,0,4,0,0,0,0,-1,4,-2,-5,1,-2,2,0,-2,2,0,0,-2,0,0,3,2,-4,0,7,0,0,0,-6,2,0,0,-4,0,0,
%X A106406 -1,6,0,0,2,-5,1,0,-2,0,2,0,0,0,0,2,0,-6,-2,4,-2,6,0,-2,0,3,0,-4,0,0,0
%N A106406 Expansion of (eta(q)eta(q^15))^2/(eta(q^3)eta(q^5)) in powers of q.
%F A106406 Euler transform of period 15 sequence [ -2, -2, -1, -2, -1, -1, -2, -2, -1, -1, -2, -1, -2, -2, -2, ...].
%F A106406 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=-v^3+4uvw+2uw^2+u^2w.
%F A106406 G.f.: Sum_{k>0} kronecker(k, 3) x^k(1-x^k)(1-x^(2k))/(1-x^(5k)) = Sum_{k>0} kronecker(k, 5) x^k(1-x^k)/(1-x^(3k)).
%F A106406 a(n) is multiplicative with a(3^e) = a(5^e) = (-1)^e, a(p^e) = (1+(-1)^e)/2 if p == 7, 11, 13, 14 (mod 15), a(p^e) = e+1 if p == 1, 4 (mod 15), a(p^e) = (e+1)(-1)^e if p == 2, 8 (mod 15). - Michael Somos Oct 19 2005
%F A106406 a(15n+7)=a(15n+11)=a(15n+13)=a(15n+14)=0.
%F A106406 G.f.: (1/2)(Sum_{n,m} x^(n^2+nm+4m^2) -x^(2n^2+nm+2m^2)). - Michael Somos Aug 25 2006
%F A106406 G.f.: x*Product_{k>0} ((1-x^k)(1-x^(15k)))^2/((1-x^(3k))(1-x^(5k))).
%F A106406 A035175(n)=|a(n)|. a(n)>0 iff n in A028957. a(n)<0 iff n in A028955.
%e A106406 q - 2*q^2 - q^3 + 3*q^4 - q^5 + 2*q^6 - 4*q^8 + q^9 + 2*q^10 +...
%o A106406 (PARI) {a(n)=local(A); if(n<1,0,n--; A=x*O(x^n); polcoeff( eta(x+A)^2*eta(x^15+A)^2/(eta(x^3+A)*eta(x^5+A)),n))}
%o A106406 (PARI) a(n)=if(n<1, 0, sumdiv(n,d, kronecker(d,3)*kronecker(n/d,5)))
%o A106406 (PARI) {a(n)=local(A,p,e,x); if(n<1, 0, A=factor(n); prod(k=1,matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==3|p==5, (-1)^e, if((p%15)!=2^(x=valuation(p%15,2)), (e+1)%2, (e+1)*(-1)^(x*e))))))}
%o A106406 (PARI) {a(n)=if(n<1, 0, (qfrep([2, 1;1, 8],n, 1)-qfrep([4, 1;1, 4], n, 1))[n])} /* Michael Somos Aug 25 2006 */
%Y A106406 Adjacent sequences: A106403 A106404 A106405 this_sequence A106407 A106408 A106409
%Y A106406 Sequence in context: A129761 A123864 A035175 this_sequence A092412 A078734 A028293
%K A106406 sign,mult
%O A106406 1,2
%A A106406 Michael Somos, May 02 2005

%I A092412
%S A092412 1,2,1,3,1,2,1,0,1,2,1,3,1,2,1,1,1,2,1,3,1,2,1,0,1,2,1,3,1,2,1,2,1,2,1,
%T A092412 3,1,2,1,0,1,2,1,3,1,2,1,1,1,2,1,3,1,2,1,0,1,2,1,3,1,2,1,3,1,2,1,3,1,2,
%U A092412 1,0,1,2,1,3,1,2,1,1,1,2,1,3,1,2,1,0,1,2,1,3,1,2,1,2,1,2,1,3,1,2,1,0
%N A092412 Fixed point of the morphism 0->11, 1->12, 2->13, 3->10, starting from a(1) = 1.
%F A092412 a(n) = A001511(n) mod 4 . a(2n+1) = 1; a(2n) = a(n) + 1 mod 4 . a(n) = A035263(n) mod 2; a(n) = A033485(n) mod 2.
%t A092412 Nest[ Function[ l, {Flatten[(l /. {0 -> {1, 1}, 1 -> {1, 2}, 2 -> {1, 3}, 3 -> {1, 0}})] }], {0}, 7] (from Robert G. Wilson v Mar 04 2005)
%Y A092412 Adjacent sequences: A092409 A092410 A092411 this_sequence A092413 A092414 A092415
%Y A092412 Sequence in context: A123864 A035175 A106406 this_sequence A078734 A028293 A092782
%K A092412 easy,nonn
%O A092412 1,2
%A A092412 DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 22 2004

%I A078734
%S A078734 1,2,1,3,1,2,1,1,1,2,1,3,1,2,1,2,1,2,1,3,1,2,1,1,1,2,1,3,1,2,1,3,1,2,1,
%T A078734 3,1,2,1,1,1,2,1,3,1,2,1,2,1,2,1,3,1,2,1,1,1,2,1,3,1,2,1,1,1,2,1,3,1,2,
%U A078734 1,1,1,2,1,3,1,2,1,2,1,2,1,3,1,2,1,1,1,2,1,3,1,2,1,3,1,2,1,3,1,2,1,1,1
%N A078734 Start with 1,2, concatenate 2^k previous terms and change last term as follows : 1->2, 2->3, 3->1.
%F A078734 Sum(k=1, n, a(k))/n -> 1.57.....
%e A078734 Concatenate the 2 first terms 1,2 -> 1,2,1,2 change 2->3 gives the 4 first terms : 1,2,1,3. Concatenate those 4 first terms ->1,2,1,3,1,2,1,3 change 3->1 gives the 8 first terms : 1,2,1,3,1,2,1,1
%Y A078734 Cf. A056832, A035263.
%Y A078734 Adjacent sequences: A078731 A078732 A078733 this_sequence A078735 A078736 A078737
%Y A078734 Sequence in context: A035175 A106406 A092412 this_sequence A028293 A092782 A089242
%K A078734 nonn
%O A078734 1,2
%A A078734 Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 21 2002

%I A028293
%S A028293 1,2,1,3,1,2,1,1,2,1,3,1,2,1,1,2,1,3,1,2,1,1,2,1,3,1,2,
%T A028293 1,1,2,1,3,1,2,1,1,2,1,3,1,2,1,1,2,1,3,1,2,1,1,2,1,3,1,
%U A028293 2,1,1,2,1,3,1,2,1,1,2,1,3,1,2,1,1,2,1,3,1,2,1,1,2,1
%N A028293 Period 7.
%F A028293 a(n)=1/147*{11*(n mod 7)+32*[(n+1) mod 7]-10*[(n+2) mod 7]+53*[(n+3) mod 7]-31*[(n+4) mod 7]+32*[(n+5) mod 7]-10*[(n+6) mod 7]} with n>=0 - Paolo P. Lava (ppl(AT)spl.at), Nov 27 2006
%Y A028293 Adjacent sequences: A028290 A028291 A028292 this_sequence A028294 A028295 A028296
%Y A028293 Sequence in context: A106406 A092412 A078734 this_sequence A092782 A089242 A029423
%K A028293 nonn,easy
%O A028293 0,2
%A A028293 njas

%I A092782
%S A092782 1,2,1,3,1,2,1,1,2,1,3,1,2,1,2,1,3,1,2,1,1,2,1,3,1,2,1,3,1,2,1,1,2,1,3,
%T A092782 1,2,1,2,1,3,1,2,1,1,2,1,3,1,2,1,1,2,1,3,1,2,1,2,1,3,1,2,1,1,2,1,3,1,2,
%U A092782 1,3,1,2,1,1,2,1,3,1,2,1,2,1,3,1,2,1,1,2,1,3,1,2,1,2,1,3,1,2,1,1,2,1,3
%N A092782 Fixed point of the morphism 1 -> 12, 2 -> 13, 3 -> 1, starting from a(1) = 1.
%C A092782 See A103269 for another version with further references and comments.
%D A092782 V. F. Sirvent, Semigroups and the self-similar structure of the flipped tribonacci substitution, Applied Math. Letters, 12 (1999), 25-29. [Contains many further references.]
%F A092782 a(n) = 1 for n in A003144; a(n) = 2 for n in A003145; a(n) = 3 for n in A003146.
%t A092782 Nest[ Function[ l, {Flatten[(l /. {1 -> {1, 2}, 2 -> {1, 3}, 3 -> {1}})] }], {1}, 8] (from Robert G. Wilson v Mar 04 2005)
%Y A092782 Cf. A003144 A003145 A003146 A100619 A103269.
%Y A092782 Adjacent sequences: A092779 A092780 A092781 this_sequence A092783 A092784 A092785
%Y A092782 Sequence in context: A092412 A078734 A028293 this_sequence A089242 A029423 A059130
%K A092782 easy,nonn
%O A092782 1,2
%A A092782 DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Apr 23 2004

%I A089242
%S A089242 1,2,1,3,1,2,1,2,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,2,1,2,1,3,1,2,1,2,1,2,1,
%T A089242 3,1,2,1,2,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,2,1,2,1,3,1,2,1,3,1,2,1,3,1,2,
%U A089242 1,2,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,2,1,2,1,3,1,2,1,2,1,2,1,3,1,2,1,2,1
%N A089242 Sequence is S(infinity), where S(1) = 1, S(m+1) = concatenation S(m), a(m)+1, S(m) and a(m) is the m-th term of S(m). a(m) is also the m-th term of the sequence.
%C A089242 S(m) has 2^m - 1 elements and is palindromic for all m.
%C A089242 First occurrence of k: 1,2,4,16,65536,...,. A014221: a(n+1) = 2^a(n). This is an Ackermann function. - Robert G. Wilson v May 30 2006.
%H A089242 Robert G. Wilson v, <a href="http://www.research.att.com/~njas/sequences/b089242.txt">Table of n, a(n) for n = 1..65536</a>
%F A089242 a(m) = number of c's such that 0 = c(c(c(c(...c(m)...)))), where 2^c(n) is the highest power of 2 which divides evenly into n (i.e. a(m) = 1 + a(c(m))); also c(m) = A007814(m).
%F A089242 In other words, a(n) = number of iterates of A007814 until a zero is encountered.
%t A089242 c[n_] := (i++; Block[{k = 0, m = n}, While[ EvenQ[m], k++; m /= 2]; k]); f[n_] := (i = 0; NestWhile[c, n, # >= 1 &]; i); Array[f, 105] (from Robert G. Wilson v (rgwv(at)rgwv.com), May 30 2006)
%Y A089242 Cf. A007814.
%Y A089242 Adjacent sequences: A089239 A089240 A089241 this_sequence A089243 A089244 A089245
%Y A089242 Sequence in context: A078734 A028293 A092782 this_sequence A029423 A059130 A094959
%K A089242 nonn,easy
%O A089242 1,2
%A A089242 Leroy Quet (qq-quet(AT)mindspring.com), Dec 13 2003
%E A089242 More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Aug 31 2005

%I A029423
%S A029423 1,0,0,0,0,0,1,0,1,1,1,0,1,0,1,1,2,1,3,1,2,1,2,1,4,2,4,
%T A029423 3,4,2,5,2,5,4,6,4,8,4,7,5,8,5,10,6,10,8,11,7,13,8,13,10,
%U A029423 14,10,17,11,17,13,18,13,21,14,21,17,23,17,26,18,26,21
%N A029423 Expansion of 1/((1-x^6)(1-x^8)(1-x^9)(1-x^10)).
%Y A029423 Adjacent sequences: A029420 A029421 A029422 this_sequence A029424 A029425 A029426
%Y A029423 Sequence in context: A028293 A092782 A089242 this_sequence A059130 A094959 A108103
%K A029423 nonn
%O A029423 0,17
%A A029423 njas

%I A059130
%S A059130 1,2,1,3,1,2,1,2,3,2,4,2,3,2,1,2,1,3,1,2,1,3,4,3,5,3,4,3,1,2,1,3,1,2,1,
%T A059130 2,3,2,4,2,3,2,1,2,1,3,1,2,1,4,5,4,6,4,5,4,1,2,1,3,1,2,1,2,3,2,4,2,3,2,
%U A059130 1,2,1,3,1,2,1,3,4,3,5,3,4,3,1,2,1,3,1,2,1,2,3,2,4,2,3,2,1,2,1,3,1,2,1
%N A059130 A hierarchical sequence (W2{3}* - see A059126).
%H A059130 J. Wallgren, <a href="http://www.research.att.com/~njas/sequences/A059126.txt">Hierarchical sequences</a>
%Y A059130 Adjacent sequences: A059127 A059128 A059129 this_sequence A059131 A059132 A059133
%Y A059130 Sequence in context: A092782 A089242 A029423 this_sequence A094959 A108103 A111376
%K A059130 easy,nonn
%O A059130 0,2
%A A059130 Jonas Wallgren (jonwa(AT)ida.liu.se), Jan 19 2001

%I A094959
%S A094959 1,2,1,3,1,2,1,2,3,4,3,5,2,2,1,5,1,6,3,4,5,4,1,6,4,2,5,9,7,2,1,9,8,8,10,
%T A094959 17,10,8,7,11,5,10,7,7,10,4,3,12,8,7,12,8,1,10,12,18,18,11,10,14,10,2,1,
%U A094959 16,19,22,14,12,15,17,9,22,14,10,19,15,9,16,9,2,27,23,18,26,25,20,14,22
%N A094959 Number of positive integer coefficients in n-th Bernoulli polynomial.
%C A094959 This is, more explicitly, the number of positive integers of the form C(n+1,i)*B(i) where B(i) is the i-th Bernoulli number and C(n,k) is the binomial coefficient (k -sets from n distinct elements). The floor((n-1)/2) zero cases are excluded from this sequence. - Olivier GERARD (ogerard(AT)ext.jussieu.fr), Oct 19 2005
%D A094959 R. L. Graham et al., Concrete Math., Chapter 6.5, Bernoulli numbers
%e A094959 B(5,x)=x^5 - (5/2)*x^4 +( 5/3)*x^3 +0*x^2- (1/6)*x+0 hence a(5)=1
%t A094959 Table[Count[ DeleteCases[ Table[Binomial[j + 1, i]*BernoulliB[ i], {i, 0, j}], 0], _Integer], {j, 0, 200}] (Gerard)
%o A094959 (PARI) B(n,x)=sum(i=0,n,binomial(n,i)*bernfrac(i)*x^(n-i)); a(n)=sum(i=0,n,if(frac(polcoeff(B(n,x),i)),0,1))-floor((n-1)/2)
%Y A094959 Cf. A027641, A027642.
%Y A094959 Adjacent sequences: A094956 A094957 A094958 this_sequence A094960 A094961 A094962
%Y A094959 Sequence in context: A089242 A029423 A059130 this_sequence A108103 A111376 A001511
%K A094959 nonn
%O A094959 1,2
%A A094959 Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 19 2004

%I A108103
%S A108103 1,2,1,3,1,2,1,3,1,3,1,2,1,3,1,2,1,3,1,3,1,2,1,3,1,3,1,2,1,3,1,2,1,3,1,
%T A108103 3,1,2,1,3,1,2,1,3,1,3,1,2,1,3,1,3,1,2,1,3,1,2,1,3,1,3,1,2,1,3,1,3,1,2,
%U A108103 1,3,1,2,1,3,1,3,1,2,1,3,1,2,1,3,1,3,1,2,1,3,1,3,1,2,1,3,1,2,1,3,1,3,1
%N A108103 A Fibonacci like substitution for three-symbol substitution with characteristic polynomial: x^3-2*x-1.
%C A108103 This sequence gives a three-symbol substitution for Benoit Cloture's A066983 Real Salem Roots: {{x -> -1.}, {x -> -0.618034}, {x -> 1.61803}}
%F A108103 1->{3}, 2->{1}, 3->{1, 2, 1}
%t A108103 s[1] = {3}; s[2] = {1}; s[3] = {1, 2, 1}; t[a_] := Flatten[s /@ a]; p[0] = {1}; p[1] = t[p[0]]; p[n_] := t[p[n - 1]] a = p[12]
%Y A108103 Cf. A000045, A066983.
%Y A108103 Adjacent sequences: A108100 A108101 A108102 this_sequence A108104 A108105 A108106
%Y A108103 Sequence in context: A029423 A059130 A094959 this_sequence A111376 A001511 A065704
%K A108103 nonn,uned
%O A108103 0,2
%A A108103 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jun 03 2005

%I A111376
%S A111376 1,1,2,1,3,1,2,1,3,1,3,1,4,2,1,1,1,3,1,4,6,1,1,4,5,3,6,4,9,4,5,0,3,4,4,
%T A111376 18,1,3,4,7,0,3,25,1,5,11,4,12,7,32,11,15,15,4,24,21,27,21,31,24,17,41,
%U A111376 31,4,38,50,18,36,46,41,36,45,67,12,57,50,38,95,51,73,14,82,32,27,171,44
%V A111376 1,1,2,1,3,1,2,-1,3,1,3,1,4,2,-1,-1,1,3,1,4,6,1,-1,-4,5,-3,6,4,9,-4,-5,0,-3,4,4,
%W A111376 18,1,-3,-4,-7,0,-3,25,1,5,-11,-4,-12,-7,32,11,15,-15,4,-24,-21,27,21,31,-24,17,-41,
%X A111376 -31,4,38,50,-18,36,-46,-41,-36,45,67,-12,57,-50,-38,-95,51,73,14,82,-32,-27,-171,44
%N A111376 Let qf(a,q) = Product(1-a*q^j,j=0..infinity); g.f. is qf(q^3,q^7)*qf(q^5,q^7)*qf(q^6,q^7)/(qf(q,q^7)*qf(q^2,q^7)*qf(q^4,q^7)).
%F A111376 Euler transform of period 7 sequence [1, 1, -1, 1, -1, -1, 0, ...]. - Michael Somos Nov 11 2005
%F A111376 G.f.: Product_{k>0} (1-x^(7k-4))(1-x^(7k-2))(1-x^(7k-1))/((1-x^(7k-3))*(1-x^(7k-5))(1-x^(7k-6))) . - Michael Somos Nov 11 2005
%o A111376 (PARI) {a(n)=if(n<0, 0, polcoeff( prod(k=1,n, (1-x^k)^-kronecker(-7,k), 1+x*O(x^n)), n))} /* Michael Somos Nov 11 2005 */
%Y A111376 Cf. A111375.
%Y A111376 Adjacent sequences: A111373 A111374 A111375 this_sequence A111377 A111378 A111379
%Y A111376 Sequence in context: A059130 A094959 A108103 this_sequence A001511 A065704 A026100
%K A111376 sign
%O A111376 0,3
%A A111376 njas, Nov 09 2005

%I A001511 M0127 N0051
%S A001511 1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,1,2,1,
%T A001511 3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,7,1,2,1,3,1,2,
%U A001511 1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,1,2,1,3,1,2,1,4,1
%N A001511 The ruler function: 2^a(n) divides 2n. Or, a(n) = 2-adic valuation of 2n.
%C A001511 a(n) is the number of digits that must be counted from right to left to reach the first 1 in the binary representation of n. For example, a(12)=3 digits must be counted from right to left to reach the first 1 in 1100, the binary representation of 12. - anon, May 17 2002
%C A001511 If you are counting in binary, and the least significant bit is numbered 1, the next bit is 2, etc., a(n) is the bit that is incremented when increasing from n-1 to n. - Jud McCranie, Apr 26, 2004
%C A001511 Number of steps to reach an integer starting with (n+1)/2 and using the map x -> x*ceiling(x) (cf. A073524).
%C A001511 a(n) = number of disk to be moved at n-th step of optimal solution to Tower of Hanoi problem (comment from Andreas M. Hinz (hinz(AT)appl-math.tu-muenchen.de)).
%C A001511 Shows which bit to flip when creating the binary reflected Gray code (bits are numbered from the right, offset is 1). This is essentially equivalent to Hinz's comment. - Adam Kertesz (adamkertesz(AT)worldnet.att.net), Jul 28 2001
%C A001511 a(n) is the Hamming distance between n and n-1 (in binary). This is equivalent to Kertesz's comments above. - Tak-Shing Chan (chan12(AT)alumni.usc.edu), Feb 25 2003
%C A001511 Let S(0) = {1}, S(n) = {S(n-1), S(n-1)-{x}, x+1} where x = last term of S(n-1); sequence gives S(infinity). - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 14 2003
%C A001511 The sum of all terms up to and including the first occurrence of m is 2^m-1. - Donald Sampson (marsquo(AT)hotmail.com), Dec 01 2003
%C A001511 m appears every 2^m terms starting with the 2^(m-1)th term. - Donald Sampson (marsquo(AT)hotmail.com), Dec 08 2003
%C A001511 Sequence read mod 4 gives A092412. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 28 2004
%C A001511 If q = 2n/2^A001511(n) and if b(m) is defined by b(0)=q-1 and b(m)=2*b(m-1)+1, then 2n = b(A001511(n)) + 1. - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Dec 18 2004
%C A001511 Repeating pattern ABACABADABACABAE ... - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com), Jan 16 2005
%C A001511 Relation to C(n) = Collatz function iteration using only odd steps: a(n) is the number of right bits set in binary representation of A004767(n) (numbers of the form 4*m+3). So for m=A004767(n) it follows that there are exactly a(n) recursive steps where m<C(m). - Lambert Klasen (lambert.klasen(AT)gmx.de), Jan 23 2005
%C A001511 The ordinal transform of a sequence b_0, b_1, b_2, ... is the sequence a_0, a_1, a_2, ... where a_n is the number of times b_n has occurred in {b_0 ... b_n}.
%C A001511 Between every two instances of any positive integer m there are exactly m distinct values (1 through m-1 and one value greater than m). - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Sep 18 2006
%C A001511 Number of divisors of n of the form 2^k. - Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Jul 25 2007
%D A001511 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
%D A001511 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 2nd ed., 2001-2003; see Dim- and Dim+ on p. 98; Dividing Rulers, on pp. 436-437; The Ruler Game, pp. 469-470; Ruler Fours, Fives, ... Fifteens on p. 470.
%D A001511 Flajolet, P., Raoult, J.-C., and Vuillemin, J.; The number of registers required for evaluating arithmetic expressions. Theoret. Comput. Sci. 9 (1979), no. 1, 99-125.
%D A001511 F. Q. Gouvea, p-Adic Numbers, Springer-Verlag, 1993; see p. 23.
%D A001511 A. M. Hinz, The Tower of Hanoi, in Algebras and combinatorics (Hong Kong, 1997), 277-289, Springer, Singapore, 1999.
%D A001511 Problem 636, Math. Mag., 40 (1967), 164-165.
%D A001511 Andrew Schloss, "Towers of Hanoi" composition, in The Digital Domain. Elektra/Asylum Records 9 60303-2, 1983. Works by Jaffe (Finale to ``Silicon Valley Breakdown''), McNabb (``Love in the Asylum''), Schloss (``Towers of Hanoi''), Mattox (``Shaman''), Rush, Moorer (``Lions are Growing'') and others.
%D A001511 Dinesh Thakur, J. Number Theory, 1991.
%H A001511 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b001511.txt">Table of n, a(n) for n = 1..10000</a>
%H A001511 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A001511 J.-P. Allouche and J. Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/Papers/as0.ps">The ring of k-regular sequences</a>, Theoretical Computer Sci., 98 (1992), 163-197.
%H A001511 J. Britton, <a href="http://ccins.camosun.bc.ca/~jbritton/jbhanoi.htm">Tower of Hanoi Solution</a>
%H A001511 J. C. Lagarias and N. J. A. Sloane, Approximate squaring (<a href="http://www.research.att.com/~njas/doc/apsq.pdf">pdf</a>, <a href="http://www.research.att.com/~njas/doc/apsq.ps">ps</a>), Experimental Math., 13 (2004), 113-128.
%H A001511 Michael Naylor, <a href="http://www.ac.wwu.edu/~mnaylor/abacaba/abacaba.html">Abacaba-Dabacaba</a>
%H A001511 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0307027">Divide-and-conquer generating functions. I. Elementary sequences</a>
%H A001511 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/somedcgf.html">Some divide-and-conquer sequences ...</a>
%H A001511 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/a079944.ps">Table of generating functions</a>
%H A001511 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinaryCarrySequence.html">Link to a section of The World of Mathematics.</a>
%H A001511 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RulerFunction.html">Link to a section of The World of Mathematics.</a>
%H A001511 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TowersofHanoi.html">Link to a section of The World of Mathematics.</a>
%H A001511 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A001511 <a href="http://www.research.att.com/~njas/sequences/Sindx_Bi.html#binary">Index entries for sequences related to binary expansion of n</a>
%F A001511 a(2n+1) = 1; a(2n) = 1 + a(n). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 08 2003
%F A001511 a(n) = 2-A000120(n)+A000120(n-1), n >= 1 - from Daniele Parisse (daniele.parisse(AT)m.dasa.de).
%F A001511 a(n) = 1 + lg(abs(A003188(n)-A003188(n-1))), where lg is the base-2 logarithm.
%F A001511 Multiplicative with a(p^e) = e+1 if p = 2; 1 if p > 2. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01 2001.
%F A001511 For any real x > 1/2: lim N ->inf (1/N)*sum(n=1, N, x^(-a(n)))= 1/(2x-1); also lim N ->inf (1/N)*Sum(n=1, N, 1/a(n))=ln(2). - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 16 2001
%F A001511 s(n) = sum(k=1, n, a(k)) is asymptotic to 2*n since s(n)=2n-A000120(n). - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 31 2002
%F A001511 For any n>=0, for any m>=1, a(2^m*n+2^(m-1)) = m. - Benoit Cloitre, Nov 24 2002
%F A001511 a(n) = sum_{d divides n and d is odd} mu(d)*tau(n/d). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 04 2002
%F A001511 G.f.: A(x) = sum_{k=0..infinity} x^(2^k)/(1-x^(2^k)). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Dec 24 2002
%F A001511 a(1) = 1; for n > 1, a(n) = a(n-1)+(-1)^n*a(floor(n/2)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 25 2003
%F A001511 Sum(k = 1 through n) a(k) = 2n - number of 1's in binary representation of n. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 15 2003
%F A001511 A fixed point of the mapping 1->12; 2->13; 3->14; 4->15; 5->16; .... - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 13 2003
%F A001511 Product_{k>0}(1+x^k)^a(k) is g.f. for A000041(). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 26 2004
%F A001511 G.f. A(x) satisfies A(x) = A(x^2) + x/(1-x). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Feb 09 2006
%F A001511 a(A118413(n,k))=A002260(n,k); = a(A118416(n,k))=A002024(n,k); a(A014480(n))=A003602(A014480(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 27 2006
%F A001511 Ordinal transform of A003602. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Aug 28 2006
%F A001511 Could be extended to n <= 0 using a(-n)=a(n), a(0)=0, a(2n)=a(n)+1 unless n=0. - Michael Somos Sep 30 2006
%F A001511 Sequence = A129360 * A000005 = M*V, where M = an infinite lower triangular matrix and V = d(n) as a vector: [1, 2, 2, 3, 2, 4,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 15 2007
%F A001511 A001511 = row sums of triangle A130093. - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 13 2007
%F A001511 Dirichlet g.f.: zeta(s)*2^s/(2^s-1). - Ralf Stephan, Jun 17 2007
%F A001511 a(n)=sum_{d divides n} mu(2d)*tau(n/d). - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 21 2007
%F A001511 G.f.: x/(1-x) = Sum_{n>=1} a(n)*x^n*( 1 - x^n ) . - Paul D. Hanna (pauldhanna(AT)juno.com), Jun 22 2007
%e A001511 For example, 2^1|2, 2^2|4, 2^1|6, 2^3|8, 2^1|10, 2^2|12, ... giving the initial terms 1, 2, 1, 3, 1, 2, ...
%p A001511 A001511 := n->2-wt(n)+wt(n-1); # where wt is defined in A000120
%t A001511 Array[ If[ Mod[ #, 2] == 0, FactorInteger[ # ][[1, 2]], 0] &, 105] + 1 (* or *)
%t A001511 Nest[ Flatten[ # /. a_Integer -> {1, a + 1}] &, {1}, 7] (from Robert G. Wilson v Mar 04 2005)
%o A001511 (PARI) a(n) = sum(k=0,floor(log(n)/log(2)),floor(n/2^k)-floor((n-1)/2^k)) (from R. Stephan)
%o A001511 (PARI) a(n)=if(n%2,1,factor(n)[1,2]+1) - Jon Perry (perry(AT)globalnet.co.uk), Jun 06 2004
%o A001511 (PARI) {a(n)=if(n, valuation(n, 2)+1, 0)} /* Michael Somos Sep 30 2006 */
%o A001511 (PARI) {a(n)=if(n==1,1,polcoeff(x-sum(k=1, n-1, a(k)*x^k*(1-x^k)*(1-x+x*O(x^n))), n))} - Paul D. Hanna (pauldhanna(AT)juno.com), Jun 22 2007
%Y A001511 a(n) = A007814(n)+1, column 1 of table A050600. Cf. A018238. Sequence read mod 2 gives A035263.
%Y A001511 From Marc LeBrun: A005187(n) = Sum a(k), k <= n.
%Y A001511 Cf. A003188, A065176, A050603, A007814, A007949, A005187, A085058, A089080.
%Y A001511 Sequence is bisection of A007814, A050603, A050604, A067029, A089309.
%Y A001511 A085058 is a bisection.
%Y A001511 Cf. A003278, A117303, A000005, A129360, A130093.
%Y A001511 A094267(2n)=A050603(2n)=A050603(2n+1)=a(n). - Michael Somos Sep 30 206.
%Y A001511 This is Guy Steele's sequence GS(4,2) (see A135416).
%Y A001511 Adjacent sequences: A001508 A001509 A001510 this_sequence A001512 A001513 A001514
%Y A001511 Sequence in context: A094959 A108103 A111376 this_sequence A065704 A026100 A059127
%K A001511 mult,nonn,nice,easy,core
%O A001511 1,2
%A A001511 njas

%I A065704
%S A065704 1,2,1,3,1,2,1,4,2,2,1,3,1,2,1,5,1,4,1,3,1,2,1,4,2,2,2,3,1,2,1,6,1,2,1,
%T A065704 6,1,2,1,4,1,2,1,3,2,2,1,5,2,4,1,3,1,4,1,4,1,2,1,3,1,2,2,7,1,2,1,3,1,2,
%U A065704 1,8,1,2,2,3,1,2,1,5,3,2,1,3,1,2,1,4,1,4,1,3,1,2,1,6,1,4,2,6,1,2,1,4,1
%N A065704 Number or squares or twice squares dividing n.
%F A065704 1/2*Sum_{ d divides n } (1-(-1)^sigma(d)). Multiplicative with a(2^e) = e+1 and a(p^e) = floor(e/2)+1 for an odd prime p.
%e A065704 divisors(36) = {1, 2, 3, 4, 6, 9, 12, 18, 36}, thus a(36) = #{1, 2, 4, 9, 18, 36}=6. a(36) = 1/2*(tau(36)-((-1)^sigma(1)+(-1)^sigma(2)+(-1)^sigma(3)+(-1)^sigma(4)+(-1)^sigma(6)+(-1)^sigma(9)+(-1)^sigma(12)+(-1)^sigma(18)+(-1)^sigma(36))) = 1/2*(9-(-3)) = 6. a(36) = a(2^2*3^2) = a(2^2)*a(3^2) = (2+1)*(1+1) = 6.
%Y A065704 Cf. A000203, A028982, A046951.
%Y A065704 Adjacent sequences: A065701 A065702 A065703 this_sequence A065705 A065706 A065707
%Y A065704 Sequence in context: A108103 A111376 A001511 this_sequence A026100 A059127 A105609
%K A065704 mult,nonn
%O A065704 1,2
%A A065704 Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 04 2001
%E A065704 More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Sep 09 2002

%I A026100
%S A026100 1,2,1,3,1,2,1,4,3,2,1,4,1,2,3,5,1,5,1,4,3,2,1,6,5,2,7,4,1,6,1,6,3,2,5,8,1,2,3,7,1,
%T A026100 6,1,4,8,2,1,9,7,9,3,4,1,10,5,8,3,2,1,10,1,2,9,7,5,6,1,4,3,10,1,11,1,2,11,4,7,6,1,
%U A026100 12,12,2,1,11,5,2,3,8,1,13,7,4,3,2,5,13,1,12,9,14,1,6,1,8,13,2,1,14,1,10,3,14,1,6,5
%N A026100 a(n) = number of the column of A026098 that contains n.
%Y A026100 Adjacent sequences: A026097 A026098 A026099 this_sequence A026101 A026102 A026103
%Y A026100 Sequence in context: A111376 A001511 A065704 this_sequence A059127 A105609 A101872
%K A026100 nonn
%O A026100 1,2
%A A026100 Clark Kimberling (ck6(AT)evansville.edu)

%I A059127
%S A059127 1,2,1,3,1,2,1,4,5,4,6,4,5,4,1,2,1,3,1,2,1,7,8,7,9,7,8,7,1,2,1,3,1,2,1,
%T A059127 4,5,4,6,4,5,4,1,2,1,3,1,2,1,10,11,10,12,10,11,10,1,2,1,3,1,2,1,4,5,4,
%U A059127 6,4,5,4,1,2,1,3,1,2,1,7,8,7,9,7,8,7,1,2,1,3,1,2,1,4,5,4,6,4,5,4,1,2,1
%N A059127 A hierarchical sequence (W2{3} - see A059126).
%H A059127 J. Wallgren, <a href="http://www.research.att.com/~njas/sequences/A059126.txt">Hierarchical sequences</a>
%Y A059127 Adjacent sequences: A059124 A059125 A059126 this_sequence A059128 A059129 A059130
%Y A059127 Sequence in context: A001511 A065704 A026100 this_sequence A105609 A101872 A069929
%K A059127 easy,nonn
%O A059127 0,2
%A A059127 Jonas Wallgren (jonwa(AT)ida.liu.se), Jan 19 2001

%I A105609
%S A105609 1,0,1,2,1,3,1,2,1,5,1,1,1,7,1,2,1,3,1,1,1,11,1,1,1,13,1,1,1,1,1,2,1,17,
%T A105609 1,1,1,19,1,1
%V A105609 1,0,-1,-2,1,-3,-1,2,-1,5,-1,1,1,-7,1,2,1,-3,-1,1,1,-11,-1,1,1,13,-1,1,1,1,-1,2,1,17,1,
%W A105609 1,1,-19,1,1
%N A105609 Sylvester numbers for 1/(1+x^2).
%H A105609 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SylvesterCyclotomicNumber.html">Sylvester Cyclotomic Number</a>.
%F A105609 a(n)=product{k=1..n-1, if(gcd(n, k)=1, (I+I*exp(2*pi*I*k/n), 1)}, I=sqrt(-1)
%e A105609 (x+I)(x-I)=1+x^2
%t A105609 f[n_] := FullSimplify[ Expand[Times @@ (I + I*Exp[2Pi*I*Select[Range[n], GCD[ #, n] == 1 &]/n])]]; Table[ f[n], {n, 0, 32}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 02 2005)
%Y A105609 Cf. A020513, A105608, A105607.
%Y A105609 Adjacent sequences: A105606 A105607 A105608 this_sequence A105610 A105611 A105612
%Y A105609 Sequence in context: A065704 A026100 A059127 this_sequence A101872 A069929 A101312
%K A105609 easy,sign,more
%O A105609 0,4
%A A105609 Paul Barry (pbarry(AT)wit.ie), Apr 15 2005

%I A101872
%S A101872 1,2,1,3,1,2,1,5,2,2,1,3,1,2,1,7,1,4,1,3,1,2,1,5,2,2,3,3,1,2,1,11,1,2,1,
%T A101872 6,1,2,1,5,1,2,1,3,2,2,1,7,2,4,1,3,1,6,1,5,1,2,1,3,1,2,2,15,1,2,1,3,1,2,
%U A101872 1,10,1,2,2,3,1,2,1,7,5,2,1,3,1,2,1,5,1,4,1,3,1,2,1,11,1,4,2,6,1,2,1,5
%N A101872 Number of abelian groups of order 2n.
%Y A101872 Bisection of A000688.
%Y A101872 Adjacent sequences: A101869 A101870 A101871 this_sequence A101873 A101874 A101875
%Y A101872 Sequence in context: A026100 A059127 A105609 this_sequence A069929 A101312 A035942
%K A101872 nonn,easy
%O A101872 1,2
%A A101872 njas, Jan 28 2005
%E A101872 More terms from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 10 2006

%I A069929
%S A069929 1,1,2,1,3,1,2,2,2,1,3,1,2,2,2,1,4,1,3,2,2,1,3,1,2,3,3,1,3,1,3,2,2,1,3,
%T A069929 1,2,2,2,1,4,1,2,2,2,1,5,1,3,2,2,1,3,1,3,2,2,1,5,1,3,2,2,2,3,1,2,3,4,1,
%U A069929 3,1,2,2,4,1,3,1,2,2,2,1,5,1,2,2,3,1,4,1,2,2,2,1,3,1,2,2,2,1,4,1,4,2,2
%N A069929 Number of k, 1<=k<=n, such that k^3+1 divides n^3+1.
%F A069929 Conjecture : (1/n)*sum(k=1, n, a(k)) = C*ln(ln(n))+o(ln(ln(n)) with 1<C<3/2
%o A069929 (PARI) for(n=1,150,print1(sum(i=1,n,if((n^3+1)%(i^3+1),0,1)),","))
%Y A069929 Cf. A066743.
%Y A069929 Adjacent sequences: A069926 A069927 A069928 this_sequence A069930 A069931 A069932
%Y A069929 Sequence in context: A059127 A105609 A101872 this_sequence A101312 A035942 A036989
%K A069929 easy,nonn
%O A069929 1,3
%A A069929 Benoit Cloitre (benoit7848c(AT)orange.fr), May 05 2002

%I A101312
%S A101312 2,1,3,1,2,2,2,2,1,1,2,2,1,3,1,1,2,2,1,2,1,2,2,1,3,1,1,3,2,1,3,1,2,2,2,
%T A101312 2,1,1,2,2,1,3,1,1,2,2,1,2,1,2,2,1,3,1,1,3,2,1,3,1,2,2,2,2,1,1,2,2,1,3,
%U A101312 1,1,2,2,1,2,1,2,2,1,3,1,1,3,2,1,3,1,2,2,2,2,1,1,2,2,1,3,1,1,2,2,1,2,1
%N A101312 Number of "Friday the 13ths" in year n (starting at 1901).
%C A101312 This sequence is basically periodic with period 28 [example: a(1901) = a(1929) = a(1957)], with "jumps" when it passes a non-leap-year century such as 2100 [all centuries which are not multiples of 400]. At these points [for example, a(2101)], the sequence simply "jumps" to a different point in the same pattern, "dropping back" 12 entries [or equivalently, "skipping ahead" 16 entries], but still continuing with the same repeating [period 28] pattern. Every year has at least 1 "Friday the 13th," and no year has more than 3. On average, 3 of every 7 years (43%) have 1 "Friday the 13th," 3 of every 7 years (43%) have 2 of them, and only 1 in 7 years (14%) has 3 of them. Conjecture: The same basic repeating pattern results if we seek the number of "Sunday the 22nds" or "Wednesday the 8ths" or anything else similar, with the only difference being that the sequence starts at a different point in the repeating pattern.
%e A101312 a(2004) = 2, since there were 2 "Friday the 13ths" that year: Feb 13, 2004 and Aug 13, 2004 and both fell on a Friday.
%t A101312 (*Load <<Miscellaneous`Calendar` package first*) s={};For[n=1901, n<=2200, t=0;For[m=1, m<=12, If[DayOfWeek[{n, m, 13}]===Friday, t++ ];m++ ];AppendTo[s, t];n++ ];s
%Y A101312 Adjacent sequences: A101309 A101310 A101311 this_sequence A101313 A101314 A101315
%Y A101312 Sequence in context: A105609 A101872 A069929 this_sequence A035942 A036989 A035197
%K A101312 nonn
%O A101312 1901,1
%A A101312 Adam M. Kalman (mocha(AT)clarityconnect.com), Dec 22 2004

%I A035942
%S A035942 1,0,0,1,0,1,0,1,1,1,1,1,1,1,2,1,3,1,2,2,3,3,3,3,3,4,4,6,5,5,3,6,8,7,8,
%T A035942 7,7,8,10,12,11,10,10,12,15,18,15,15,15,17,20,24,24,20,20,25,29,32,32,
%U A035942 28,28,32,40,44,44,39,39,43,56,58,55,54,53,58,71,80,77,69,69,79,92
%N A035942 Number of partitions of n into distinct prime power parts (1 and 2 excluded).
%e A035942 a(16) = 3 because we can write 16=3+13=5+11=3^2+7
%Y A035942 Cf. A051613.
%Y A035942 Adjacent sequences: A035939 A035940 A035941 this_sequence A035943 A035944 A035945
%Y A035942 Sequence in context: A101872 A069929 A101312 this_sequence A036989 A035197 A091948
%K A035942 nonn
%O A035942 0,15
%A A035942 Vladeta Jovovic (vladeta(AT)Eunet.yu)

%I A036989
%S A036989 1,2,1,3,1,2,2,4,1,2,1,3,1,3,3,5,1,2,1,3,1,2,2,4,1,2,2,4,2,4,4,6,1,2,1,
%T A036989 3,1,2,2,4,1,2,1,3,1,3,3,5,1,2,1,3,1,3,3,5,1,3,3,5,3,5,5,7,1,2,1,3,1,2,
%U A036989 2,4,1,2,1,3,1,3,3,5,1,2,1,3,1,2,2,4,1,2,2,4,2,4,4,6,1,2,1,3,1,2,2,4,1
%N A036989 Read binary expansion of n from the right; keep track of the excess of 1's over 0's that have been seen so far; sequence gives 1 + maximum(excess of 1's over 0's).
%C A036989 Associated with A036988 (Remark 4 of the reference).
%D A036989 H. Niederreiter and M. Vielhaber, Tree complexity and a doubly ..., J. Complexity, 12 (1996), 187-198.
%F A036989 a(n) = 1 iff, in the binary expansion of n, reading from right to left, the number of 1's never exceeds the number of 0's.
%F A036989 a(0) = 1, a(2n) = max(a(n) - 1, 1), a(2n+1) = a(n) + 1. - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Dec 26 2006
%F A036989 Equals inverse Moebius transform (A051731) of A010060, the Thue-Morse sequence starting with "1": (1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 13 2007
%e A036989 59 in binary is 111011, excess from right to left is 1,2,1,2,3,4, maximum is 4, so a(59) = 4.
%Y A036989 Cf. A036988, A036990, A126387.
%Y A036989 Cf. A010060.
%Y A036989 Adjacent sequences: A036986 A036987 A036988 this_sequence A036990 A036991 A036992
%Y A036989 Sequence in context: A069929 A101312 A035942 this_sequence A035197 A091948 A102862
%K A036989 nonn,easy
%O A036989 0,2
%A A036989 njas
%E A036989 Edited by Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 11 2006

%I A035197
%S A035197 1,2,1,3,1,2,2,4,1,2,2,3,0,4,1,5,2,2,0,3,2,4,0,4,1,0,1,
%T A035197 6,0,2,0,6,2,4,2,3,0,0,0,4,0,4,2,6,1,0,0,5,3,2,2,0,2,2,
%U A035197 2,8,0,0,2,3,2,0,2,7,0,4,2,6,0,4,2,4,0,0,1,0,4,0,0,5,1
%N A035197 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 15.
%o A035197 (PARI) direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
%Y A035197 Adjacent sequences: A035194 A035195 A035196 this_sequence A035198 A035199 A035200
%Y A035197 Sequence in context: A101312 A035942 A036989 this_sequence A091948 A102862 A029331
%K A035197 nonn
%O A035197 1,2
%A A035197 njas

%I A091948
%S A091948 1,2,1,3,1,2,2,4,1,3,1,3,2,4,2,5,3,3,1,5,1,4,2,4,3,5,2,6,4,4,6,6,1,7,1,
%T A091948 5,1,5,1,7,2,3,1,8,1,6,2,5,3,7,2,9,2,4,3,8,2,8,4,6,4,10,2,7,9,3,5,11,1,
%U A091948 5,3,7,1,6,1,10,1,3,4,9,2,7,1,5,1,5,2,12,2,3,2,11,1,6,8,6,9,7,2,12,3,4
%N A091948 Number of values of k, 1<=k<=n, satisfying A002487(k) = A002487(n).
%Y A091948 Adjacent sequences: A091945 A091946 A091947 this_sequence A091949 A091950 A091951
%Y A091948 Sequence in context: A035942 A036989 A035197 this_sequence A102862 A029331 A037269
%K A091948 nonn
%O A091948 1,2
%A A091948 Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 11 2004

%I A102862
%S A102862 1,1,2,1,3,1,2,2,4,2,6,1,3,1,4,2,5,2,4,3,4,2,3,3,4,4,5,2,5,4,5,2,4,2,3,
%T A102862 2,5,2,3,5,3,3,3,2,4,2,3,2,5,3,4,6,4,2,2,3,4,2,2,1,3,3,3,1,2,2,3,3,3,3,
%U A102862 5,2,3,4,2,3,4,3,6,4,3,2,6,4,5,3,5,4,4,3,5,2,2,2,6,1,4,3,5,1,5,1,7,2,5
%N A102862 Numbers of prime factors of the sum of the first n primes.
%e A102862 a(3)=2 because 2+3+5=10=2*5;
%e A102862 a(5)=3 because 2+3+5+7+11=28=2*2*7.
%o A102862 (PARI) for(n=1,105,print1(bigomega(sum(k=1,n,prime(k))),",")) (Shepherd)
%Y A102862 Adjacent sequences: A102859 A102860 A102861 this_sequence A102863 A102864 A102865
%Y A102862 Sequence in context: A036989 A035197 A091948 this_sequence A029331 A037269 A124579
%K A102862 nonn
%O A102862 1,3
%A A102862 Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Mar 01 2005
%E A102862 More terms from Rick L. Shepherd (rshepherd2(AT)hotmail.com), May 17 2005

%I A029331
%S A029331 1,0,0,0,1,1,1,0,1,1,2,1,3,1,2,2,4,3,4,2,5,4,6,4,8,5,7,
%T A029331 6,10,8,11,7,12,10,14,11,17,12,16,14,21,17,22,16,24,21,
%U A029331 27,22,31,24,31,27,37,31,39,31,42,37,46,39,52,42,52,46
%N A029331 Expansion of 1/((1-x^4)(1-x^5)(1-x^6)(1-x^12)).
%Y A029331 Adjacent sequences: A029328 A029329 A029330 this_sequence A029332 A029333 A029334
%Y A029331 Sequence in context: A035197 A091948 A102862 this_sequence A037269 A124579 A035306
%K A029331 nonn
%O A029331 0,11
%A A029331 njas

%I A037269
%S A037269 1,0,0,0,1,1,1,0,1,1,2,1,3,1,2,2,4,3,4,2,5,4,6,4,8,5,7,
%T A037269 6,10,8,12,7,12,10,15,12,18,12,17,15,23,18,25,17,26,23,
%U A037269 31,25,35,26,36,31,43,35,47,36,49,43,56,47,63,49,64,56
%N A037269 Expansion of (1+x^30)/((1-x^4)(1-x^5)(1-x^6)(1-x^12)).
%D A037269 J. Igusa, On Siegel modular forms of genus 2 (II), Amer. J. Math., 86 (1964), 392-412, esp. p. 402.
%Y A037269 Adjacent sequences: A037266 A037267 A037268 this_sequence A037270 A037271 A037272
%Y A037269 Sequence in context: A091948 A102862 A029331 this_sequence A124579 A035306 A101691
%K A037269 nonn
%O A037269 0,11
%A A037269 njas

%I A124579
%S A124579 0,0,1,0,2,1,3,1,2,2,4,3,5,3,4,4,6,5,7,6,5,6,8,7,8,7,9,10,9,10,11,11,8,
%T A124579 9,10,12,12,11,12,13,13,14,15,14,15,13,16,16,17,18,14,19,17,20,15,21,16,
%U A124579 17,18,22,19,18,23,24,19,20,21,25,20,22,23,26,24,21,27,28,22,25,26,29
%N A124579 Number of natural numbers less than n which have identical Moebius mu values.
%e A124579 a(10) = 2 because in the first nine integers less than 10, mu(1) = mu(6) = mu(10).
%t A124579 a[n_] := a[n] = MoebiusMu@n; f[n_] := Count[a /@ Range[n - 1], a@n]; Array[f, 80]
%Y A124579 Cf. A124580.
%Y A124579 Cf. A008683, A002321.
%Y A124579 Adjacent sequences: A124576 A124577 A124578 this_sequence A124580 A124581 A124582
%Y A124579 Sequence in context: A102862 A029331 A037269 this_sequence A035306 A101691 A070094
%K A124579 nonn
%O A124579 1,5
%A A124579 Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 05 2006

%I A035306
%S A035306 1,1,2,1,3,1,2,2,5,1,2,1,3,1,7,1,2,3,3,2,2,1,5,1,11,1,2,2,3,1,13,1,2,
%T A035306 1,7,1,3,1,5,1,2,4,17,1,2,1,3,2,19,1,2,2,5,1,3,1,7,1,2,1,11,1,23,1,2,
%U A035306 3,3,1,5,2,2,1,13,1,3,3,2,2,7,1,29,1,2,1,3,1,5,1,31,1,2,5,3,1,11,1,2
%N A035306 List factors of each number in order (each prime factor is followed by its power).
%e A035306 1={1,1}, 2={2,1}, 3={3,1}, 4={2,2}, 5={5,1}, 6={2,1,3,1}, ...
%t A035306 Flatten[ Array[ FactorInteger[ # ]&, 40 ] ]
%o A035306 (MAGMA) [ Factorization(n) : n in [1..120]];
%Y A035306 Adjacent sequences: A035303 A035304 A035305 this_sequence A035307 A035308 A035309
%Y A035306 Sequence in context: A029331 A037269 A124579 this_sequence A101691 A070094 A105497
%K A035306 nonn,tabf
%O A035306 1,3
%A A035306 njas

%I A101691
%S A101691 1,1,2,1,3,1,2,2,5,1,2,2,4,2,4,4,9,1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,17,1,2,
%T A101691 2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,33,1,2,
%U A101691 2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8
%N A101691 A modular binomial sum sequence.
%C A101691 a(2^n)=A094373(n). a(2^n-1)=1,1,1,2,4,8,16,...
%F A101691 a(n)=sum{k=0..n, mod(binomial(2n-2, k), 2)}
%Y A101691 Cf. A048896.
%Y A101691 Adjacent sequences: A101688 A101689 A101690 this_sequence A101692 A101693 A101694
%Y A101691 Sequence in context: A037269 A124579 A035306 this_sequence A070094 A105497 A132662
%K A101691 easy,nonn
%O A101691 0,3
%A A101691 Paul Barry (pbarry(AT)wit.ie), Dec 11 2004

%I A070094
%S A070094 0,0,1,0,1,0,1,1,1,1,2,1,3,1,2,2,5,2,5,3,3,4,6,3,6,4,7,6,10,4,10,7,8,7,
%T A070094 10,7,14,8,12,8,17,10,17,12,13,14,20,12,21,14,18,16,25,15,23,18,22,20,
%U A070094 30,16,32,21,29,23,32,21,38,27,33,26,43,25
%N A070094 Number of acute integer triangles with perimeter n and relatively prime side lengths.
%C A070094 a(n) = A051493(n) - A070102(n) - A070109(n).
%H A070094 R. Zumkeller, <a href="http://www.research.att.com/~njas/sequences/a070080.txt">Integer-sided triangles</a>
%e A070094 For n=10 there are A005044(10) = 2 integer triangles: [2,4,4] and [3,3,4]; both are acute, but GCD(2,4,4)>1, therefore a(9) = 1.
%Y A070094 Cf. A070080, A070081, A070082, A070093, A070096, A070099, A070084, A070119.
%Y A070094 Adjacent sequences: A070091 A070092 A070093 this_sequence A070095 A070096 A070097
%Y A070094 Sequence in context: A124579 A035306 A101691 this_sequence A105497 A132662 A132589
%K A070094 nonn
%O A070094 1,11
%A A070094 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 05 2002

%I A105497
%S A105497 2,1,3,1,2,3,1,2,3,1,3,2,1,2,3,1,2,3,1,3,2,1,2,3,1,2,3,1,2,3,1,2,3,2,1,
%T A105497 3,1,3,2,1,2,3,1,2,3,1,3,2,1,2,3,1,2,3,1,2,3,1,2,3,2,1,3,1,3,2,1,2,3,1,
%U A105497 2,3,1,3,2,1,2,3,1,2,3,1,3,2,1,2,3,1,2,3,1,3,2,1,2,3,1,2,3,2,1,3,1,2,3
%N A105497 A simple "Fractal Jump Sequence" (FJS).
%C A105497 Successive trigrams have different digits, starting with the lowest available one -- as one can see here: 2 1 3 1 2 3 1 2 3 1 3 2 1 2 3 1 2 3 1 3 2 1 2 3 1 2 3 1 2 3 1 2 3 gives: [213] [123] [123] [132] [123] [123] [132] [123] [123] [123] [123]...
%Y A105497 Adjacent sequences: A105494 A105495 A105496 this_sequence A105498 A105499 A105500
%Y A105497 Sequence in context: A035306 A101691 A070094 this_sequence A132662 A132589 A054843
%K A105497 base,easy,nonn
%O A105497 2,1
%A A105497 Eric Angelini (eric.angelini(AT)kntv.be), May 02 2005

%I A132662
%S A132662 0,1,1,1,2,1,3,1,2,3,1,4,1,2,3,5,1,3,5,1,6,5,4,1,5,3,7,1,2,3,4,1,5,7,6,
%T A132662 5,1,2,3,9,4,7,5,1,9,6,7,10,1,2,8,4,9,1,11,10,7,9,5,8,1,2,3,4,12,5,11,1,
%U A132662 9,7,11,5,8,1,13,9,7,11,5,10,13,6,1,3,11,14,13,12,1,11,7,4,13,10,5,15
%N A132662 Values of x such that x^2+y^2=n where x<=y for values of n given by A008784.
%Y A132662 Cf. A008784 A132663.
%Y A132662 Adjacent sequences: A132659 A132660 A132661 this_sequence A132663 A132664 A132665
%Y A132662 Sequence in context: A101691 A070094 A105497 this_sequence A132589 A054843 A038566
%K A132662 nonn
%O A132662 1,5
%A A132662 Colin Barker (c.barker(AT)orange.fr), Aug 24 2007

%I A132589
%S A132589 1,1,1,2,1,3,1,2,3,2,1,1,1,2,3,4,1,6,1,1,1,1,1,2,1,1,1,4
%N A132589 Let c(k) be the k-th term of the flattened irregular array where the m-th row contains the positive integers that are <= m and are coprime to m. (c(k) = A038566(k).) Then a(n) = GCD(c(n),n).
%e A132589 A038566: 1,1,1,2,1,3,1,2,3,4,1,5,1,2,...
%e A132589 The 14th term of this list is 2.
%e A132589 So a(14) = GCD(2,14) = 2.
%Y A132589 Cf. A132587, A132588, A038566.
%Y A132589 Adjacent sequences: A132586 A132587 A132588 this_sequence A132590 A132591 A132592
%Y A132589 Sequence in context: A070094 A105497 A132662 this_sequence A054843 A038566 A020652
%K A132589 more,nonn
%O A132589 1,4
%A A132589 Leroy Quet (qq-quet(AT)mindspring.com), Aug 23 2007

%I A054843
%S A054843 2,1,3,1,2,3,2,1,3,3,2,2,2,2,5,1,2,3,2,2,5,2,2,2,3,2,4,3,2,4,2,1,4,2,4,
%T A054843 4,2,2,4,2,2,4,2,2,7,2,2,2,3,3,4,2,2,4,5,2,4,2,2,4,2,2,6,1,4,5,2,2,4,4,
%U A054843 2,3,2,2,6,2,4,5,2,2,5,2,2,4,4,2,4,2,2,6,5,2,4,2,4,2,2,3,6,3,2,4,2,2,9
%N A054843 Number of sequences of consecutive nonnegative integers (including sequences of length 1) that sum to n.
%F A054843 a(n) = A001227(n) + A010054(n). G.f.: Sum_{k>0} x^(k*(k-1)/2)/(1-x^k). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 10 2004
%e A054843 a(1) = 2 because 1 = 0+1 or 1; a(15) = 5 because 15 = 0+1+2+3+4+5 or 1+2+3+4+5 or 4+5+6 or 7+8 or 15.
%Y A054843 Cf. A001227, A054844.
%Y A054843 Adjacent sequences: A054840 A054841 A054842 this_sequence A054844 A054845 A054846
%Y A054843 Sequence in context: A105497 A132662 A132589 this_sequence A038566 A020652 A096107
%K A054843 easy,nonn
%O A054843 1,1
%A A054843 Henry Bottomley (se16(AT)btinternet.com), Apr 13 2000

%I A038566
%S A038566 1,1,1,2,1,3,1,2,3,4,1,5,1,2,3,4,5,6,1,3,5,7,1,2,4,5,7,8,1,3,7,9,1,2,3,
%T A038566 4,5,6,7,8,9,10,1,5,7,11,1,2,3,4,5,6,7,8,9,10,11,12,1,3,5,9,11,13,1,2,
%U A038566 4,7,8,11,13,14,1,3,5,7,9,11,13,15,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15
%N A038566 Numerators in canonical bijection from positive integers to positive rationals <= 1: arrange fractions by increasing denominator then by increasing numerator:
%C A038566 Also numerators in canonical bijection from positive integers to all positive rational numbers: arrange fractions in triangle in which n-th row the phi(n) contains fractions i/j with GCD(i,j) = 1, i+j=n, i=1,...,n-1, j=n-1,...,1. Denominators (A020653) are obtained by reversing each row.
%C A038566 Also triangle in which n-th row gives phi(n) numbers between 1 and n that are relatively prime to n.
%D A038566 Richard Courant and Herbert Robbins. What Is Mathematics?, Oxford, 1941, pp. 79-80.
%D A038566 H. Lauwerier, Fractals, Princeton Univ. Press, p. 23.
%H A038566 David Wasserman, <a href="http://www.research.att.com/~njas/sequences/b038566.txt">Table of n, a(n) for n = 0..100000</a>
%H A038566 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A038566.text">Rows of rationals, n=1..24.</a>
%H A038566 <a href="http://www.research.att.com/~njas/sequences/Sindx_St.html#Stern">Index entries for sequences related to Stern's sequences</a>
%H A038566 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A038566 The n-th "clump" consists of the phi(n) integers <= n and prime to n.
%e A038566 The beginning of the list of positive rationals <= 1: 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, ... (this is A038566/A038567).
%e A038566 The beginning of the triangle giving all positive rationals: 1/1; 1/2, 2/1; 1/3, 3/1; 1/4, 2/3, 3/2, 4/1; 1/5, 5/1; 1/6, 2/5, 3/4, 4/3, 5/2, 6/1; .. (this is A038566/A020653).
%e A038566 The beginning of the triangle in which n-th row gives numbers between 1 and n that are relatively prime to n: 1; 1; 1,2; 1,3; 1,2,3,4; 1,5; ...
%p A038566 s := proc(n) local i,j,k,ans; i := 0; ans := [ ]; for j while i<n do for k to j do if gcd(j,k) = 1 then ans := [ op(ans),k ]; i := i+1 fi od od; RETURN(ans); end; s(100);
%t A038566 Flatten[Table[Flatten[Position[GCD[Table[Mod[j, w], {j, 1, w-1}], w], 1]], {w, 1, 100}], 2]
%Y A038566 Cf. A020652, A020653, A038566-A038569, A000010, A002088, A060837, A071970.
%Y A038566 A054424 gives mapping to Stern-Brocot tree.
%Y A038566 Row sums give rationals A111992(n)/A069220(n), n>=1.
%Y A038566 Adjacent sequences: A038563 A038564 A038565 this_sequence A038567 A038568 A038569
%Y A038566 Sequence in context: A132662 A132589 A054843 this_sequence A020652 A096107 A128487
%K A038566 nonn,frac,core,nice,tabf
%O A038566 0,4
%A A038566 njas
%E A038566 More terms from Erich Friedman (erich.friedman(AT)stetson.edu).

%I A020652
%S A020652 1,1,2,1,3,1,2,3,4,1,5,1,2,3,4,5,6,1,3,5,7,1,2,4,5,7,8,1,3,7,9,1,2,3,4,5,
%T A020652 6,7,8,9,10,1,5,7,11,1,2,3,4,5,6,7,8,9,10,11,12,1,3,5,9,11,13,1,2,4,7,8,
%U A020652 11,13,14,1,3,5,7,9,11,13,15,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,1,5
%N A020652 Numerators in canonical bijection from positive integers to positive rationals.
%D A020652 Richard Courant and Herbert Robbins. What Is Mathematics?, Oxford, 1941, pp. 79-80.
%D A020652 H. Lauwerier, Fractals, Princeton Univ. Press, p. 23.
%H A020652 David Wasserman, <a href="http://www.research.att.com/~njas/sequences/b020652.txt">Table of n, a(n) for n = 1..100000</a>
%H A020652 <a href="http://www.research.att.com/~njas/sequences/Sindx_St.html#Stern">Index entries for sequences related to Stern's sequences</a>
%H A020652 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%e A020652 ..., 1/8, 2/7, 4/5, 5/4, 7/2, 8/1, ...
%p A020652 with (numtheory): A020652 := proc (n) local sum, j, k; sum := 0: k := 2: while (sum < n) do: sum := sum + phi(k): k := k + 1: od: sum := sum - phi(k-1): j := 1; while sum < n do: if gcd(j,k-1) = 1 then sum := sum + 1: fi: j := j+1: od: RETURN (j-1): end: # from UlrSchimke(AT)aol.com, Nov 06, 2001
%Y A020652 Essentially the same as A038566, which is the main entry for this sequence.
%Y A020652 Cf. A020653, A038567-A038569.
%Y A020652 A054424 gives mapping to Stern-Brocot tree.
%Y A020652 Adjacent sequences: A020649 A020650 A020651 this_sequence A020653 A020654 A020655
%Y A020652 Sequence in context: A132589 A054843 A038566 this_sequence A096107 A128487 A056609
%K A020652 nonn,frac,core,nice
%O A020652 1,3
%A A020652 David W. Wilson (davidwwilson(AT)comcast.net)

%I A096107
%S A096107 1,1,2,1,3,1,2,3,4,1,5,1,6,1,3,5,7,1,8,1,3,7,9,1,2,3,4,5,6,7,8,9,10,1,5,
%T A096107 7,11,1,5,8,12,1,13,1,2,4,7,8,11,13,14,1,3,5,7,9,11,13,15,1,2,3,4,5,6,7,
%U A096107 8,9,10,11,12,13,14,15,16,1,17,1,7,8,11,12,18,1,3,7,9,11,13,17,19,1,8
%N A096107 Triangle read by rows: row n lists cubic residues modulo n.
%o A096107 (PARI) maybecubegcd1(n) = { for(x=2,n, b=floor(x-1); a=vector(b+1); for(y=1,b, z=y^3%x; if(z<>0, a[y]=z; ) ); s=vecsort(a); c=1; for(j=2,b+1, if(s[j]<>s[j-1], c++; if(gcd(s[j],x)==1,print1(s[j]",")) ) ); ) }
%Y A096107 Cf. A096087.
%Y A096107 Adjacent sequences: A096104 A096105 A096106 this_sequence A096108 A096109 A096110
%Y A096107 Sequence in context: A054843 A038566 A020652 this_sequence A128487 A056609 A014673
%K A096107 nonn,tabl
%O A096107 2,3
%A A096107 Cino Hilliard (hillcino368(AT)gmail.com), Jul 22 2004
%E A096107 Edited by Don Reble (djr(AT)nk.ca), May 07 2006

%I A128487
%S A128487 1,1,2,1,3,1,2,3,4,2,3,4,1,2,3,4,5,6,1,3,5,7,1,2,4,5,7,8,2,4,5,6,8,1,2,
%T A128487 3,4,5,6,7,8,9,10,2,3,4,8,9,10,1,2,3,4,5,6,7,8,9,10,11,12,2,4,6,7,8,10,
%U A128487 12,3,5,6,9,10,12,1,3,5,7,9,11,13,15,1,2,3,4,5,6,7,8,9,10,11,12,13,14
%N A128487 Irregular array where n-th row is the positive integers < n which are coprime to exactly one distinct prime divisor of n.
%C A128487 Number of terms in n-th row is A126080(n). Row 1 has zero terms, so the first listed row is row 2.
%e A128487 Concerning row 12: 1,5,7,11 don't appear because they are each coprime to 2 AND 3 (the distinct prime divisors of 12). 6 doesn't appear because it is coprime to neither prime dividing 12. The row consists of 2,3,4,8,9,10 because each term is coprime to exactly one prime divisor of 12 (ie, is coprime to 2 or 3, but not to both).
%Y A128487 Cf. A126080, A128488.
%Y A128487 Adjacent sequences: A128484 A128485 A128486 this_sequence A128488 A128489 A128490
%Y A128487 Sequence in context: A038566 A020652 A096107 this_sequence A056609 A014673 A085392
%K A128487 nonn,tabf
%O A128487 2,3
%A A128487 Leroy Quet (qq-quet(AT)mindspring.com), Mar 04 2007
%E A128487 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 08 2007

%I A056609
%S A056609 1,1,2,1,3,1,2,3,5,1,1,1,7,5,2,1,3,1,5,7,11,1,1,5,13,3,7,1,1,1,2,11,17,
%T A056609 7,1,1,19,13,1,1,7,1,11,1,23,1,1,7,5,17,13,1,3,11,1,19,29,1,1,1,31,1,2,
%U A056609 13,11,1,17,23,1,1,1,1,37,5,19,11,13,1,1,3,41,1,1,17,43,29,11,1,1,13
%N A056609 Quotient of square-free kernels of A002944(n) and A001405.
%e A056609 The quotient is 1, i.e. LCM[..,n]=LCM[..,C(n,i),..] if n=Prime[k]-1 (at values of A006093).
%Y A056609 A002944, A001405, A003418, A002110.
%Y A056609 Adjacent sequences: A056606 A056607 A056608 this_sequence A056610 A056611 A056612
%Y A056609 Sequence in context: A020652 A096107 A128487 this_sequence A014673 A085392 A089384
%K A056609 nonn
%O A056609 1,3
%A A056609 Labos E. (labos(AT)ana.sote.hu), Aug 07 2000

%I A014673
%S A014673 1,1,1,2,1,3,1,2,3,5,1,2,1,7,5,2,1,3,1,2,7,11,1,2,5,13,3,2,1,3,1,2,11,
%T A014673 17,7,2,1,19,13,2,1,3,1,2,3,23,1,2,7,5,17,2,1,3,11,2,19,29,1,2,1,31,3,2,
%U A014673 13,3,1,2,23,5,1,2,1,37,5,2,11,3,1,2,3,41,1,2,17,43,29,2,1,3,13,2,31,47
%N A014673 Smallest prime factor of greatest proper divisor of n.
%C A014673 a(n) = A020639(A032742(n));
%C A014673 For n>1: a(n) = 1 iff n is prime; a(A001358(n)) = A084127(n); a(A000961(n)) = A020639(A000961(n)).
%F A014673 A117357(n) = A020639(A054576(n)); A117358(n) = A032742(A054576(n)) = A054576(n)/A117357(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 10 2006
%t A014673 PrimeFactors[ n_ ] := Flatten[ Table[ # [ [ 1 ] ], {1} ] & /@ FactorInteger[ n ] ]; f[ n_ ] := Block[ {gpd = Divisors[ n ][ [ -2 ] ]}, If[ gpd == 1, 1, PrimeFactors[ gpd ][ [ 1 ] ] ] ]; Table[ If[ n == 1, 1, f[ n ] ], {n, 1, 95} ]
%Y A014673 Cf. A085392, A085393.
%Y A014673 Adjacent sequences: A014670 A014671 A014672 this_sequence A014674 A014675 A014676
%Y A014673 Sequence in context: A096107 A128487 A056609 this_sequence A085392 A089384 A057059
%K A014673 nonn
%O A014673 1,4
%A A014673 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 24 2003

%I A085392
%S A085392 1,1,1,2,1,3,1,2,3,5,1,3,1,7,5,2,1,3,1,5,7,11,1,3,5,13,3,7,1,5,1,2,11,
%T A085392 17,7,3,1,19,13,5,1,7,1,11,5,23,1,3,7,5,17,13,1,3,11,7,19,29,1,5,1,31,7,
%U A085392 2,13,11,1,17,23,7,1,3,1,37,5,19,11,13,1,5,3,41,1,7,17,43,29,11,1,5,13
%N A085392 Largest prime factor of greatest proper divisor of n.
%t A085392 PrimeFactors[n_] := Flatten[ Table[ # [[1]], {1}] & /@ FactorInteger[n]]; f[n_] := Block[{gpd = Divisors[n][[ -2]]}, If[gpd == 1, 1, PrimeFactors[gpd][[ -1]] ]]; Table[ If[n == 1, 1, f[n]], {n, 1, 95}]
%Y A085392 Cf. A014673, A085393.
%Y A085392 Adjacent sequences: A085389 A085390 A085391 this_sequence A085393 A085394 A085395
%Y A085392 Sequence in context: A128487 A056609 A014673 this_sequence A089384 A057059 A027750
%K A085392 nonn
%O A085392 1,4
%A A085392 Robert G. Wilson v (rgwv(AT)rgwv.com) and Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 26 2003

%I A089384
%S A089384 1,1,1,2,1,3,1,2,3,5,1,6,1,7,5,2,1,6,1,10,7,11,1,6,5,13,3,14,1,15,1,2,
%T A089384 11,17,7,6,1,19,13,10,1,21,1,22,15,23,1,6,7,10,17,26,1,6,11,14,19,29,1,
%U A089384 30,1,31,21,2,13,33,1,34,23,35,1,6,1,37,15,38,11,39,1,10,3,41,1,42
%N A089384 Greatest square-free proper divisor of n, a(1) = 1.
%C A089384 a(n) = if n is square-free then A032742(n) else A007947(n);
%C A089384 a(n) = A008966(n)*A032742(n) + (1-A008966(n))*A007947(n).
%Y A089384 Cf. A005117, A013929.
%Y A089384 Adjacent sequences: A089381 A089382 A089383 this_sequence A089385 A089386 A089387
%Y A089384 Sequence in context: A056609 A014673 A085392 this_sequence A057059 A027750 A087295
%K A089384 nonn
%O A089384 1,4
%A A089384 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Dec 28 2003

%I A057059
%S A057059 1,2,1,3,1,2,4,1,3,2,5,1,4,2,3,6,1,5,2,4,3,7,1,6,2,5,3,4,8,1,7,2,6,3,5,
%T A057059 4,9,1,8,2,7,3,6,4,5,10,1,9,2,8,3,7,4,6,5,11,1,10,2,9,3,8,4,7,5,6,12,1,
%U A057059 11,2,10,3,9,4,8,5,7,6,13,1,12,2,11,3,10
%N A057059 Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; ... Define i(m) and j(m) by R(i(m),j(m)) = m. Then a(n) = j(A057027(n)).
%C A057059 Since A057027 is a permutation of the natural numbers, every natural number occurs in this sequence infinitely many times.
%Y A057059 Cf. A057058.
%Y A057059 Adjacent sequences: A057056 A057057 A057058 this_sequence A057060 A057061 A057062
%Y A057059 Sequence in context: A014673 A085392 A089384 this_sequence A027750 A087295 A130517
%K A057059 nonn
%O A057059 1,2
%A A057059 Clark Kimberling (ck6(AT)evansville.edu), Jul 30 2000

%I A027750
%S A027750 1,1,2,1,3,1,2,4,1,5,1,2,3,6,1,7,1,2,4,8,1,3,9,1,2,5,10,1,11,1,2,3,4,6,12,1,13,1,
%T A027750 2,7,14,1,3,5,15,1,2,4,8,16,1,17,1,2,3,6,9,18,1,19,1,2,4,5,10,20,1,3,7,21,1,2,11,
%U A027750 22,1,23,1,2,3,4,6,8,12,24,1,5,25,1,2,13,26,1,3,9,27,1,2,4,7,14,28,1,29
%N A027750 Triangle read by rows in which row n list the divisors of n.
%C A027750 Or, in the list of natural numbers (A000027), replace n by its divisors.
%C A027750 This gives the first elements of the ordered pairs (a,b) a >= 1, b >= 1 ordered by their product ab.
%H A027750 Franklin T. Adams-Watters, <a href="http://www.research.att.com/~njas/sequences/b027750.txt">Rows 1 ... 1000, flattened</a>
%H A027750 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Divisor.html">Divisor</a>
%F A027750 a(A006218(n-1) + k) = k-divisor of n, 1<=k<=A000005(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 10 2006
%e A027750 1; 1,2; 1,3; 1,2,4; 1,5; 1,2,3,6; ...
%t A027750 Flatten[ Table[ Flatten [ Divisors[ n ] ], {n, 1, 30} ] ]
%o A027750 (MAGMA) [Divisors(n) : n in [1..20]];
%Y A027750 Cf. A056534, A056538, A027751.
%Y A027750 Adjacent sequences: A027747 A027748 A027749 this_sequence A027751 A027752 A027753
%Y A027750 Sequence in context: A085392 A089384 A057059 this_sequence A087295 A130517 A056951
%K A027750 nonn,easy,tabf
%O A027750 1,3
%A A027750 njas
%E A027750 More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

%I A087295
%S A087295 0,0,1,2,1,3,1,2,4,2,1,3,2,5,3,2,2,3,4,3,3,6,2,4,2,3,3,3,4,5,3,4,3,4,7,
%T A087295 3,3,5,4,3,2,4,2,4,4,5,3,6,4,4,5,4,3,5,3,8,3,4,4,4,6,5,3,4,4,3,5,4,4,5,
%U A087295 4,5,3,6,4,4,7,5,4,5,4,6,5,4,3,5,6,4,4,9,3,4,5,5,4,5,4,7,5,6,4,5,3,5,4
%N A087295 Successive remainders when computing the euclidean algorithm for (n,m) where m is any positive integer having no common factor with n, gives a list ending with a sublist of Fibonacci sequence. Find m such that this sublist has the greatest length, and define a(n) as this length.
%e A087295 a(5) = 3 because computing euclidean algorithm for (5,8) gives 3, 2, 1 as successive remainders, all three belonging to Fibonacci sequence.
%Y A087295 Adjacent sequences: A087292 A087293 A087294 this_sequence A087296 A087297 A087298
%Y A087295 Sequence in context: A089384 A057059 A027750 this_sequence A130517 A056951 A130212
%K A087295 easy,nonn
%O A087295 0,4
%A A087295 Thomas Baruchel (baruchel(AT)users.sourceforge.net), Oct 19 2003

%I A130517
%S A130517 1,2,1,3,1,2,4,2,1,3,5,3,1,2,4,6,4,2,1,3,5,7,5,3,1,2,4,6,8,6,4,2,1,3,5,
%T A130517 7,9,7,5,3,1,2,4,6,8
%N A130517 A model of the atomic nucleus.
%C A130517 1 is equal to 2 protons. 2 is equal to 2+2 protons. 3 is equal to 2+2+2 protons...
%C A130517 Natural numbers represent the subshells.
%e A130517 A geometric model of the atomic nucleus:
%e A130517 ..............-------------------------------------------------
%e A130517 ..............|...-----------------------------------------...|
%e A130517 ..............|...|...---------------------------------...|...|
%e A130517 ..............|...|...|...-------------------------...|...|...|
%e A130517 ..............|...|...|...|...-----------------...|...|...|...|
%e A130517 ..............|...|...|...|...|...---------...|...|...|...|...|
%e A130517 ..............|...|...|...|...|...|...-...|...|...|...|...|...|
%e A130517 ..............i...h...g...f...d...p...s...p...d...f...g...h...i
%e A130517 ..............|...|...|...|...|...|.......|...|...|...|...|...|
%e A130517 ..............|...|...|...|...|.......1.......|...|...|...|...|
%e A130517 ..............|...|...|...|.......2.......1.......|...|...|...|
%e A130517 ..............|...|...|.......3.......1.......2.......|...|...|
%e A130517 ..............|...|.......4.......2.......1.......3.......|...|
%e A130517 ..............|.......5.......3.......1.......2.......4.......|
%e A130517 ..................6.......4.......2.......1.......3.......5....
%e A130517 ..............7.......5.......3.......1.......2.......4.......6
%e A130517 ...............................................................
%e A130517 ...........13/2.11/2.9/2.7/2.5/2.3/2.1/2.1/2.3/2.5/2.7/2.9/2.11/2
%e A130517 ..............|...|...|...|...|...|...|...|...|...|...|...|...|
%e A130517 ..............|...|...|...|...|...|...-----...|...|...|...|...|
%e A130517 ..............|...|...|...|...|...-------------...|...|...|...|
%e A130517 ..............|...|...|...|...---------------------...|...|...|
%e A130517 ..............|...|...|...-----------------------------...|...|
%e A130517 ..............|...|...-------------------------------------...|
%e A130517 ..............|...---------------------------------------------
%Y A130517 Cf. A056951, A130556.
%Y A130517 Adjacent sequences: A130514 A130515 A130516 this_sequence A130518 A130519 A130520
%Y A130517 Sequence in context: A057059 A027750 A087295 this_sequence A056951 A130212 A133737
%K A130517 nonn,tabl
%O A130517 1,2
%A A130517 Omar E. Pol (Polnucleus(AT)gmail.com), Aug 08 2007, Aug 12 2007

%I A056951
%S A056951 1,2,1,3,1,2,4,2,1,3,5,3,1,2,4,6,4,2,1,3,5,7,5,3,1,2,4,6,8,6,4,2,1,3,5,
%T A056951 7,9,7,5,3,1,2,4,6,8,10,8,6,4,2,1,3,5,7,9,11,9,7,5,3,1,2,4,6,8,10,12,
%U A056951 10,8,6,4,2,1,3,5,7,9,11,13,11,9,7,5,3,1,2,4,6,8,10,12,14,12,10,8,6,4
%V A056951 -1,-2,1,-3,-1,2,-4,-2,1,3,-5,-3,-1,2,4,-6,-4,-2,1,3,5,-7,-5,-3,-1,2,4,6,-8,-6,-4,-2,
%W A056951 1,3,5,7,-9,-7,-5,-3,-1,2,4,6,8,-10,-8,-6,-4,-2,1,3,5,7,9,-11,-9,-7,-5,-3,-1,2,4,6,8,
%X A056951 10,-12,-10,-8,-6,-4,-2,1,3,5,7,9,11,-13,-11,-9,-7,-5,-3,-1,2,4,6,8,10,12,-14,-12,-10
%N A056951 Triangle whose rows show the result of flipping the first, first two, ..., and finally first n coins when starting with the stack (1,2,3,4,...,n) [starting with all heads up, where signs show whether particular coins end up heads or tails.].
%F A056951 T(n, k) = 2k-n-b with 1<=k<=n (where b=2 if 2k<=n+1, b=1 otherwise)
%e A056951 Third row is constructed by starting from (1,2,3), going to (-1,2,3), then going to (-2,1,3) and finally going to (-3,-1,2). Rows are: (-1), (-2,1), (-3,-1,2), (-4,-2,1,3) etc. as each row is reverse of previous row, with signs changed and -n added as the first term in the row.
%Y A056951 A003558 is the number of times the operation needs to be repeated to return to the starting point, taking no account of heads/tails (i.e. signs). A002326 is the number required if heads/tails (i.e. signs) are also required to return to their original position.
%Y A056951 Adjacent sequences: A056948 A056949 A056950 this_sequence A056952 A056953 A056954
%Y A056951 Sequence in context: A027750 A087295 A130517 this_sequence A130212 A133737 A125047
%K A056951 easy,sign,tabl
%O A056951 1,2
%A A056951 Henry Bottomley (se16(AT)btinternet.com), Sep 05 2000

%I A130212
%S A130212 1,2,1,3,1,2,4,2,2,2,5,2,2,2,4,5,3,4,2,4,2,7,3,4,2,4,2,6,8,4,4,4,4,2,6,
%T A130212 4,9,4,6,4,4,2,6,4,6,10,5,6,4,8,2,6,4,6,4
%N A130212 A000012 * A054522.
%C A130212 Row sums = the triangular series, A000217: (1, 3, 6, 10, 15,...). Right border = A000010, phi(n): (1, 1, 2, 2, 4, 2, 6,...). A130211 = A054522 * A000012
%F A130212 A000012 * A054522 as infinite lower triangular matrices.
%e A130212 First few rows of the triangle are:
%e A130212 1;
%e A130212 2, 1;
%e A130212 3, 1, 2;
%e A130212 4, 2, 2, 2;
%e A130212 5, 2, 2, 2, 4;
%e A130212 6, 3, 4, 2, 4, 2;
%e A130212 7, 3, 4, 2, 4, 2, 6;
%e A130212 8, 4, 4, 4, 4, 2, 6, 4;
%e A130212 9, 4, 6, 4, 4, 2, 6, 4, 6;
%e A130212 10, 5, 6, 4, 8, 2, 6, 4, 6, 4;
%e A130212 ...
%Y A130212 Cf. A000010, A130211, A054522, A000217.
%Y A130212 Adjacent sequences: A130209 A130210 A130211 this_sequence A130213 A130214 A130215
%Y A130212 Sequence in context: A087295 A130517 A056951 this_sequence A133737 A125047 A045898
%K A130212 nonn,tabl
%O A130212 1,2
%A A130212 Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2007

%I A133737
%S A133737 1,2,1,3,1,2,4,2,2,3,5,2,2,3,6,6,3,4,3,6,7,7,3,4,3,6,7,14,8,4,4,6,6,7,
%T A133737 14,17,9,4,6,6,6,7,14,17,27,10,5,6,6,12,7,14,17,27,34
%N A133737 A000012 * A133736.
%C A133737 Row sums = A026905, partial sums of A000041, the partition numbers with offset 1, = (1, 3, 6, 11, 18, 29, 44, 66,...).
%F A133737 A000012 * A133736 as infinite lower triangular matrices.
%e A133737 First few rows of the triangle are:
%e A133737 1;
%e A133737 2, 1;
%e A133737 3, 1, 2;
%e A133737 4, 2, 2, 3;
%e A133737 5, 2, 2, 3, 6;
%e A133737 6, 3, 4, 3, 6, 7;
%e A133737 7, 3, 4, 3, 6, 7, 14;
%e A133737 ...
%Y A133737 Cf. A133736, A026905, A000041.
%Y A133737 Adjacent sequences: A133734 A133735 A133736 this_sequence A133738 A133739 A133740
%Y A133737 Sequence in context: A130517 A056951 A130212 this_sequence A125047 A045898 A036262
%K A133737 nonn,tabl
%O A133737 1,2
%A A133737 Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 22 2007

%I A125047
%S A125047 1,2,1,3,1,2,4,3,1,2,1,3,4,2,4,3,1,2,1,3,1,2,4,3,4,2,1,3,4,2,4,3,1,2,1,
%T A125047 3,1,2,4,3,1,2,1,3,4,2,4,3,4,2,1,3,1,2,4,3,4,2,1,3,4,2,4,3,1,2,1,3,1,2,
%U A125047 4,3,1,2,1,3,4,2,4,3,1,2,1,3,1,2,4,3,4,2,1,3,4,2,4,3,4,2,1,3,1,2,4,3,1
%N A125047 Infinite word generated by mapping 1->12, 2->13, 3->43, 4->42 starting at 1.
%C A125047 Infinite word over 4-letter alphabet that contains no squares in arithmetic progressions of odd difference. - Ralf Stephan, May 09 2007
%H A125047 J.-Y. Kao et al., <a href="http://arxiv.org/abs/math.CO/0608607">Words avoiding repetitions in arithmetic progressions</a>
%F A125047 Recurrence: a(1)=1, a(4n)=3, a(4n+2)=2, a(8n+3)=1, a(8n+7)=4, a(4n+1)=a(2n+1). - Ralf Stephan, May 09 2007
%e A125047 1 -> 12 -> 1213 -> 12131242 -> 1213124312134243 -> ...
%o A125047 (PARI) {a(n)=local(A); if(n<1, 0, A=[1]; while(length(A)<n, A=concat(vector(length(A), k, [[1, 2], [1, 3], [4, 3], [4, 2]][A[k]]))); A[n])}
%Y A125047 Cf. A038190.
%Y A125047 Adjacent sequences: A125044 A125045 A125046 this_sequence A125048 A125049 A125050
%Y A125047 Sequence in context: A056951 A130212 A133737 this_sequence A045898 A036262 A046924
%K A125047 nonn
%O A125047 1,2
%A A125047 Michael Somos, Nov 17 2006

%I A045898
%S A045898 1,2,1,3,1,2,4,5,1,2,1,3,1,5,4,3,1,2,1,3,1,2,4,5
%N A045898 a(n) = one of five triples of directions in n-th triple of moves in the optimal solution of the Tower of Hanoi; it is a square-free sequence over a five-letter alphabet.
%D A045898 A. M. Hinz, Square-free Tower of Hanoi sequences, Enseign. Math. (2) 42(1996), 257-264.
%D A045898 A. M. Hinz, The Tower of Hanoi, in Algebras and combinatorics (Hong Kong, 1997), 277-289, Springer, Singapore, 1999.
%Y A045898 Adjacent sequences: A045895 A045896 A045897 this_sequence A045899 A045900 A045901
%Y A045898 Sequence in context: A130212 A133737 A125047 this_sequence A036262 A046924 A108415
%K A045898 nonn,more
%O A045898 1,2
%A A045898 Andreas M. Hinz (hinz(AT)appl-math.tu-muenchen.de)

%I A036262
%S A036262 2,1,3,1,2,5,1,0,2,7,1,2,2,4,11,1,2,0,2,2,13,1,2,0,0,2,4,17,1,2,0,0,0,
%T A036262 2,2,19,1,2,0,0,0,0,2,4,23,1,2,0,0,0,0,0,2,6,29,1,0,2,2,2,2,2,2,4,2,31,
%U A036262 1,0,0,2,0,2,0,2,0,4,6,37,1,0,0,0,2,2,0,0,2,2,2,4,41,1,0,0,0,0,2,0,0,0
%N A036262 Triangle of numbers arising from Gilbreath's conjecture: successive absolute differences of primes.
%C A036262 The conjecture is that the leading term is always 1.
%C A036262 Odlyzko has checked it for primes up to pi(10^13) = 3*10^11.
%D A036262 R. K. Guy, Unsolved Problems Number Theory, A10.
%D A036262 R. B. Killgrove and K. E. Ralston, On a conjecture concerning the primes, Math.Tables Aids Comput. 13(1959), 121-122.
%D A036262 H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 208.
%D A036262 F. Proth, Sur la serie des nombres premiers, Nouv. Corresp. Math., 4 (1878) 236-240.
%D A036262 W. Sierpinski, L'induction incomplete dans la theorie des nombres, Scripta Math. 28 (1967), 5-13.