The Database of Integer Sequences, Part 3 

     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.

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(start)



%I A029300
%S A029300 1,0,0,1,0,0,2,0,1,2,0,1,4,0,2,4,1,2,6,1,4,6,2,4,10,2,6,
%T A029300 10,4,6,14,4,10,14,6,10,20,6,14,20,10,14,26,10,20,26,14,
%U A029300 20,35,14,26,35,20,26,44,20,35,44,26,35,56,26,44,56,35
%N A029300 Expansion of 1/((1-x^3)(1-x^6)(1-x^8)(1-x^12)).
%Y A029300 Adjacent sequences: A029297 A029298 A029299 this_sequence A029301 A029302 A029303
%Y A029300 Sequence in context: A058709 A025842 A058685 this_sequence A096397 A103919 A035445
%K A029300 nonn
%O A029300 0,7
%A A029300 njas

%I A096397
%S A096397 0,0,1,0,2,0,1,2,0,1,4,2,6,1,1,0,8,3,8,4,6,4,8,4,0,3,9,6,14,2,9,8,10,6,
%T A096397 10,0,18,6,6,8,20,4,21,10,12,9,18,8,0,9,14,12,26,8,11,12,18,13,26,8,30,
%U A096397 11,17,0,24,6,34,16,22,10,28,12,36,13,18,18,30,10,28,16,0,18,39,12,32
%N A096397 a(n) = #{ 1<=i<=n : k(n,i)=-1 } where k(n,i) is the Kronecker symbol.
%o A096397 (PARI) a(n)=sum(i=1,n,if(kronecker(n,i)+1,0,1))
%Y A096397 Cf. A000010, A051953.
%Y A096397 Adjacent sequences: A096394 A096395 A096396 this_sequence A096398 A096399 A096400
%Y A096397 Sequence in context: A025842 A058685 A029300 this_sequence A103919 A035445 A053603
%K A096397 nonn
%O A096397 1,5
%A A096397 Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 06 2004

%I A103919
%S A103919 1,0,1,1,0,1,0,2,0,1,2,0,2,0,1,0,4,0,2,0,1,3,0,5,0,2,0,1,0,7,0,5,0,2,0,
%T A103919 1,5,0,9,0,5,0,2,0,1,0,12,0,10,0,5,0,2,0,1,7,0,17,0,10,0,5,0,2,0,1,0,19,
%U A103919 0,19,0,10,0,5,0,2,0,1,11,0,28,0,20,0,10,0,5,0,2,0,1,0,30,0,33,0,20,0
%N A103919 Triangle of numbers of partitions of n with total number of odd parts equal to k from {1,..,n}.
%C A103919 The partition (0) of n=0 is included. For n>0 no part 0 appears.
%C A103919 The first (k=0) column gives the number of partitions without odd parts, i.e. those with even parts only. See A035363.
%C A103919 Without the alternating zeros this becomes a triangle with columns given by the rows of the S_n(m) table shown in the Riordan reference.
%D A103919 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
%H A103919 W. Lang: <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A103919.text">First 11 rows.</a>
%F A103919 a(n, k)=number of partitions of n>=0, which have exactly k odd parts (and possibly even parts) for k from {0, ..., n}.
%F A103919 sum(k*T(n,k),k=0..n)=A066897(n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 17 2006
%F A103919 G.f.=G(t,x)=1/product((1-tx^(2j-1))(1-x^(2j)), j=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 17 2006
%e A103919 [1],[0,1],[1,0,1],[0,2,0,1],[2,0,2,0,1],[0,4,0,2,0,1],...
%e A103919 a(0,0)=1 because n=0 has no odd part, only one even part, 0, by definition. a(5,3)=2 because there are two partitions (1,1,3) and (1,1,1,2) of 5 with exactly 3 odd parts.
%p A103919 g:=1/product((1-t*x^(2*j-1))*(1-x^(2*j)),j=1..20): gser:=simplify(series(g,x=0,22)): P[0]:=1: for n from 1 to 18 do P[n]:=coeff(gser,x^n) od: for n from 0 to 18 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 17 2006
%Y A103919 Row sums A000041 (partition numbers). Columns: k=0: A035363 (with zero entries) A000041 (without zero entries), k=1: A000070, k=2: A000097, k=3: A000098, k=4: A000710.
%Y A103919 Cf. A066897.
%Y A103919 Adjacent sequences: A103916 A103917 A103918 this_sequence A103920 A103921 A103922
%Y A103919 Sequence in context: A058685 A029300 A096397 this_sequence A035445 A053603 A085794
%K A103919 nonn,easy,tabl
%O A103919 0,8
%A A103919 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Mar 24 2005

%I A035445
%S A035445 0,0,0,0,1,0,0,1,0,1,0,0,2,0,1,2,0,2,0,1,4,0,2,3,1,5,0,2,7,1,5,5,2,9,1,
%T A035445 5,12,2,10,8,5,17,2,10,20,5,19,13,10,29,5,20,32,10,34,20,20,49,10,36,
%U A035445 50,20,59,32,37,78,20,64,77,37,97,50,66,124,37,107,117,67,157,79
%N A035445 Number of partitions of n into parts 8k or 8k+5.
%Y A035445 Adjacent sequences: A035442 A035443 A035444 this_sequence A035446 A035447 A035448
%Y A035445 Sequence in context: A029300 A096397 A103919 this_sequence A053603 A085794 A004548
%K A035445 nonn
%O A035445 1,13
%A A035445 Olivier Gerard (ogerard(AT)ext.jussieu.fr)

%I A053603
%S A053603 0,0,1,0,2,0,1,2,0,2,0,2,1,2,0,0,4,0,2,0,1,2,2,0,2,2,0,2,0,2,1,
%T A053603 4,0,0,2,0,2,2,2,2,0,0,3,2,0,0,4,0,2,2,0,4,0,0,0,2,3,2,2,0,2,2,
%U A053603 0,0,2,2,2,2,0,2,2,0,3,2,0,0,4,0,0,2,0,6,0,2,2,0,0,2,2,0,1,2,2
%N A053603 Number of ways to write n as an ordered sum of two nonzero triangular numbers.
%Y A053603 Cf. A000217, A007294, A051611, A051533, A052343-A052348, A053604.
%Y A053603 Adjacent sequences: A053600 A053601 A053602 this_sequence A053604 A053605 A053606
%Y A053603 Sequence in context: A096397 A103919 A035445 this_sequence A085794 A004548 A125921
%K A053603 nonn
%O A053603 0,5
%A A053603 njas, Jan 20 2000

%I A085794
%S A085794 0,1,0,2,0,1,2,0,2,1,0,1,2,0,1,0,2,0,1,2,0,2,1,0,2,0,1,0,2,1,0,1,2,0,1,
%T A085794 0,2,0,1,2,0,2,1,0,1,2,0,1,0,2,1,0,1,2,0,2,1,0,2,0,1,0,2,1,0,1,2,0,1,0,
%U A085794 2,0,1,2,0,2,1,0,1,2,1,0,2,0,1,0,2,1,0,1,2,0,1,0,2,0,1,2,0,2,1,0,2,0,1
%N A085794 Lexicographic minimal square-free infinite ternary word.
%C A085794 Apparently it is an open problem to find a good alternate characterization of this sequence.
%Y A085794 Adjacent sequences: A085791 A085792 A085793 this_sequence A085795 A085796 A085797
%Y A085794 Sequence in context: A103919 A035445 A053603 this_sequence A004548 A125921 A029299
%K A085794 nonn,easy
%O A085794 0,4
%A A085794 Claude Lenormand (claude.lenormand(AT)free.fr), Jul 24 2003; corrected Jul 25, 2003
%E A085794 More terms from John W. Layman (layman(AT)math.vt.edu), May 18 2004

%I A004548
%S A004548 1,2,0,1,2,0,2,1,2,2,2,2,1,2,1,0,2,2,1,2,1,1,2,0,1,0,1,2,2,2,1,0,1,
%T A004548 2,0,1,1,0,0,2,2,2,0,0,0,1,2,1,0,0,2,0,0,0,0,0,2,2,2,2,0,2,1,0,2,1,
%U A004548 0,2,1,1,1,0,0,2,0,0,0,0,2,0,1,0,2,2,0,2,0,2,1,2,0,0,1,2,2,0,0,2,1
%N A004548 Expansion of sqrt(3) in base 3.
%Y A004548 Adjacent sequences: A004545 A004546 A004547 this_sequence A004549 A004550 A004551
%Y A004548 Sequence in context: A035445 A053603 A085794 this_sequence A125921 A029299 A049803
%K A004548 nonn,base,cons
%O A004548 1,2
%A A004548 njas

%I A125921
%S A125921 1,2,0,1,2,0,2,1,2,3,1,2,4,5,3,0,3,5,1,2,1,3,2,1,2,1,2,5,6,2,1,5,4,0,2,
%T A125921 0,6,2,1,2,8,1,2,0,5,1,6,3,1,2,4,5,3,0,3,5,1,2,1,6,2,1,3,4,8,3,6,2,1,5,
%U A125921 4,0,3,0,6,8,1,2,8,1,2,0,5,1,6,3,1,9,4,5,4,10,3,5,9,4,1,12,2
%N A125921 Sprague-Grundy values for octal game .314.
%D A125921 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982; see Chapter 4, p. 104.
%Y A125921 Adjacent sequences: A125918 A125919 A125920 this_sequence A125922 A125923 A125924
%Y A125921 Sequence in context: A053603 A085794 A004548 this_sequence A029299 A049803 A053121
%K A125921 nonn
%O A125921 1,2
%A A125921 Richard Sabey (richardsabey(AT)hotmail.co.uk), Jan 24 2007

%I A029299
%S A029299 1,0,0,1,0,0,2,0,1,2,0,2,3,0,3,3,1,4,4,2,5,4,4,6,6,5,7,
%T A029299 7,7,8,10,8,10,12,10,12,15,11,15,17,14,18,20,16,22,22,20,
%U A029299 25,26,23,29,29,28,32,34,32,37,38,37,41,44,41,47,49,47
%N A029299 Expansion of 1/((1-x^3)(1-x^6)(1-x^8)(1-x^11)).
%Y A029299 Adjacent sequences: A029296 A029297 A029298 this_sequence A029300 A029301 A029302
%Y A029299 Sequence in context: A085794 A004548 A125921 this_sequence A049803 A053121 A113408
%K A029299 nonn
%O A029299 0,7
%A A029299 njas

%I A049803
%S A049803 0,0,0,1,2,0,1,2,0,2,4,3,5,7,6,8,10,0,2,4,3,5,7,6,8,10,0,3,6,6,9,12,12,
%T A049803 15,18,9,12,15,15,18,21,21,24,27,18,21,24,24,27,30,30,33,36,0,3,6,6,9,
%U A049803 12,12,15,18,9,12,15,15,18,21,21,24,27,18
%N A049803 a(n)=n mod 3 + n mod 9 + ... + n mod 3^k, where 3^k<=n<3^(k+1).
%Y A049803 Adjacent sequences: A049800 A049801 A049802 this_sequence A049804 A049805 A049806
%Y A049803 Sequence in context: A004548 A125921 A029299 this_sequence A053121 A113408 A022337
%K A049803 nonn
%O A049803 1,5
%A A049803 Clark Kimberling (ck6(AT)evansville.edu)

%I A053121
%S A053121 1,0,1,1,0,1,0,2,0,1,2,0,3,0,1,0,5,0,4,0,1,5,0,9,0,5,0,1,0,14,0,14,0,6,
%T A053121 0,1,14,0,28,0,20,0,7,0,1,0,42,0,48,0,27,0,8,0,1,42,0,90,0,75,0,35,0,9,
%U A053121 0,1,0,132,0,165,0,110,0,44,0,10,0,1,132,0,297,0,275,0,154,0,54,0,11,0
%N A053121 Catalan triangle (with 0's). Inverse lower triangular matrix of A049310(n,m) (coefficients of Chebyshev's S polynomials).
%C A053121 "The Catalan triangle is formed in the same manner as Pascal's triangle, except that no number may appear on the left of the vertical bar." [Conway and Smith]
%C A053121 G.f. for row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n): c(z^2)/(1-x*z*c(z^2)). Row sums (x=1): A001405 (central binomial).
%C A053121 In the language of the Shapiro et al. reference such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group. The g.f. Ginv(x) of the m=0 column of the inverse of a given Bell-matrix (here A049310)
%C A053121 is obtained from its g.f. of the m=0 column (here G(x)=1/(1+x^2)) by Ginv(x)=(f^{(-1)}(x))/x, with f(x) := x*G(x) and f^{(-1)}is the compositional inverse function of f (here one finds, with Ginv(0)=1, c(x^2)). See the Shapiro et al. reference.
%C A053121 Walks with a wall: triangle of number of n step walks from (0,0) to (n,m) where each step goes from (a,b) to (a+1,b+1) or (a+1,b-1) and the path stays in the nonnegative quadrant.
%C A053121 Row sums of squares equals the Catalan sequence (A000108); for row 6: A000108(6) = 5^2 + 0^2 + 9^2 + 0^2 + 5^2 + 0^2 + 1^2 = 132. - Paul D. Hanna (pauldhanna(AT)juno.com), Apr 23 2005
%C A053121 Number of involutions of {1,2,...,n} that avoid the patterns 132 and have exactly k fixed points. Example: T(4,2)=3 because we have 2134, 4231, and 3214. Number of involutions of {1,2,...,n} that avoid the patterns 321 and have exactly k fixed points. Example: T(4,2)=3 because we have 1243, 1324, and 2134. Number of involutions of {1,2,...,n} that avoid the patterns 213 and have exactly k fixed points. Example: T(4,2)=3 because we have 1243, 1432, and 4231. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 12 2006
%C A053121 Triangle T(n,k), 0<=k<=n, read by rows given by : T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=T(n-1,1), T(n,k)=T(n-1,k-1)+T(n-1,k+1) for k>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 30 2007
%C A053121 This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1 . Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; ((1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 25 2007
%C A053121 Riordan array (c(x^2),xc(x^2)), where c(x) is the g.f. of Catalan numbers A000108 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 25 2007
%D A053121 I. Bajunaid et al., Function series, Catalan numbers and random walks on trees, Amer. Math. Monthly 112 (2005), 765-785.
%D A053121 J. H. Conway and D. A. Smith, On Quaternions and Octonions, A K Peters, Ltd., Natick, MA, 2003. See p. 60. MR1957212 (2004a:17002)
%D A053121 E. Deutsch, A. Robertson and D. Saracino, Refined restricted involutions, European Journal of Combinatorics 28 (2007), 481-498 (see pp. 486 and 498).
%D A053121 V. E. Hoggatt, Jr. and M. Bicknell, Catalan and Related Sequences Arising from Inverses of Pascal's Triangle Matrices, Fibonacci Quart. 14 (1976) 395-405.
%D A053121 W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fibonacci Quart. 38,5 (2000) 408-419; Note 4, pp. 414-415.
%D A053121 A. Nkwanta, Lattice paths and RNA secondary structures, in African Americans in Mathematics, ed. N. Dean, Amer. Math. Soc., 1997, pp. 137-147.
%D A053121 L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.
%D A053121 W.-J. Woan, Area of Catalan Paths, Discrete Math., 226 (2001), 439-444.
%H A053121 C. Banderier and D. Merlini, <a href="http://algo.inria.fr/banderier/Papers/infjumps.ps">Lattice paths with an infinite set of jumps</a>
%H A053121 W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, <a href="http://jacobi.math.wvu.edu/~mays/Papers/apd2.ps">A Pascal rhombus</a>, Fibonacci Quarterly, 35 (1997), 318-328.
%H A053121 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A053121 a(n, m) := 0 if n<m or n-m odd, else a(n, m) = (m+1)*binomial(n+1, (n-m)/2)/(n+1);
%F A053121 a(n, m) = (4*(n-1)*a(n-2, m) + 2*(m+1)*a(n-1, m-1))/(n+m+2), a(n, m)=0 if n<m, a(n, -1) := 0, a(0, 0)=1=a(1, 1), a(1, 0)=0.
%F A053121 G.f. for m-th column: c(x^2)*(x*c(x^2))^m, where c(x) = g.f. for Catalan numbers A000108.
%F A053121 a(n, m)=a(n-1, m-1)+a(n-1, m+1) if n>0 and m >= 0, a(0, 0)=1, a(0, m)=0 if m>0, a(n, m)=0 if m<0 - Henry Bottomley (se16(AT)btinternet.com), Jan 25 2001
%F A053121 Sum_{k>=0} T(m, k)*T(n, k) = 0 if m+n is odd; Sum_{k>=0} T(m, k)*T(n, k) = A000108((m+n)/2) if m+n is even . - Philippe DELEHAM, May 26 2005
%F A053121 T(n,k)=sum{i=0..n, (-1)^(n-i)*C(n,i)*sum{j=0..i, C(i,j)*(C(i-j,j+k)-C(i-j,j+k+2))}}; Column k has e.g.f. BesselI(k,2x)-BesselI(k+2,2x); - Paul Barry (pbarry(AT)wit.ie), Feb 16 2006
%F A053121 Sum_{k, 0<=k<=n}T(n,k)*(k+1)=2^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 22 2007
%F A053121 Sum_{j, j>=0}T(n,j)*binomial(j,k)= A054336(n,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 30 2007
%F A053121 T(2*n+1,2*k+1)=A039598(n,k), T(2*n,2*k)=A039599(n,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 16 2007
%e A053121 .......|...1
%e A053121 .......|.......1
%e A053121 .......|...1.......1
%e A053121 .......|.......2.......1
%e A053121 .......|...2.......3.......1
%e A053121 .......|.......5.......4.......1
%e A053121 .......|...5.......9.......5.......1
%e A053121 .......|......14......14.......6.......1
%e A053121 .......|..14......28......20.......7.......1
%e A053121 .......|......42......48......27.......8.......1
%e A053121 {1}; {0,1}; {1,0,1}; {0,2,0,1}; {2,0,3,0,1};... E.g. fourth row corresponds to polynomial p(3,x)= 2*x+x^3.
%p A053121 T:=proc(n,k): if n+k mod 2 = 0 then (k+1)*binomial(n+1,(n-k)/2)/(n+1) else 0 fi end: for n from 0 to 13 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 12 2006
%Y A053121 Cf. A008315, A049310, A033184, A000108, A001405. Another version: A008313.
%Y A053121 Variant without zero-diagonals: A033184 and with rows reversed: A009766.
%Y A053121 Adjacent sequences: A053118 A053119 A053120 this_sequence A053122 A053123 A053124
%Y A053121 Sequence in context: A125921 A029299 A049803 this_sequence A113408 A022337 A025687
%K A053121 easy,nice,tabl,nonn
%O A053121 0,8
%A A053121 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)

%I A113408
%S A113408 1,0,1,1,0,1,0,2,0,1,2,0,3,0,1,0,6,0,4,0,1,3,0,12,0,5,0,1,0,12,0,20,0,6,
%T A113408 0,1,6,0,30,0,30,0,7,0,1,0,30,0,60,0,42,0,8,0,1,10,0,90,0,105,0,56,0,9,
%U A113408 0,1,0,60,0,210,0,168,0,72,0,10,0,1,20,0,210,0,420,0,252,0,90,0,11,0,1
%N A113408 Riordan array (1/(1-x^2-x^4*c(x^4)),x*c(x^2)), c(x) the g.f. of A000108.
%C A113408 Row sums are A113409. Diagonal sums are A005773(n+1) with interpolated zeros.
%F A113408 T(n, k)=C((n+k)/2, k)*C(floor((n-k)/2), floor((n-k)/4))(1+(-1)^(n-k))/2.
%e A113408 Triangle begins
%e A113408 1;
%e A113408 0,1;
%e A113408 1,0,1;
%e A113408 0,2,0,1;
%e A113408 2,0,3,0,1;
%e A113408 0,6,0,4,0,1;
%e A113408 3,0,12,0,5,0,1;
%Y A113408 Adjacent sequences: A113405 A113406 A113407 this_sequence A113409 A113410 A113411
%Y A113408 Sequence in context: A029299 A049803 A053121 this_sequence A022337 A025687 A025666
%K A113408 easy,nonn,tabl
%O A113408 0,8
%A A113408 Paul Barry (pbarry(AT)wit.ie), Oct 28 2005

%I A022337
%S A022337 0,0,1,0,1,2,0,1,2,0,3,1,2,0,3,1,4,2,0,3,1,4,2,0,5,3,1,4,2,0,5,3,1,6,4,2,
%T A022337 0,5,3,1,6,4,2,0,7,5,3,1,6,4,2,0,7,5,3,1,8,6,4,2,0,7,5,3,1,8,6,4,2,0,9,7,
%U A022337 5,3,1,8,6,4,2,0,9,7,5,3,1,10,8,6,4,2,0,9,7,5,3,1,10,8,6,4,2,0,11,9,7,5
%N A022337 Exponent of 5 (value of j) in n-th number of form 3^i*5^j.
%C A022337 a(n) = A112765(A003593(n)) = A112754(n) - A022336(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Sep 18 2005
%Y A022337 Cf. A112762, A022336.
%Y A022337 Adjacent sequences: A022334 A022335 A022336 this_sequence A022338 A022339 A022340
%Y A022337 Sequence in context: A049803 A053121 A113408 this_sequence A025687 A025666 A025679
%K A022337 nonn
%O A022337 1,6
%A A022337 Clark Kimberling (ck6(AT)evansville.edu)

%I A025687
%S A025687 0,0,1,0,1,2,0,1,2,0,3,1,2,0,3,1,4,2,0,3,1,4,2,0,5,3,1,4,2,0,5,3,1,6,4,2,
%T A025687 0,5,3,1,6,4,2,0,7,5,3,1,6,4,2,0,7,5,3,1,8,6,4,2,0,7,5,3,1,8,6,4,2,9,0,7,
%U A025687 5,3,1,8,6,4,2,9,0,7,5,3,10,1,8,6,4,2,9,0,7,5,3,10,1,8,6,4,11,2,9,0,7,5
%N A025687 Exponent of 10 (value of j) in n-th number of form 5^i*10^j.
%Y A025687 Adjacent sequences: A025684 A025685 A025686 this_sequence A025688 A025689 A025690
%Y A025687 Sequence in context: A053121 A113408 A022337 this_sequence A025666 A025679 A071491
%K A025687 nonn
%O A025687 1,6
%A A025687 David W. Wilson (davidwwilson(AT)comcast.net)

%I A025666
%S A025666 0,0,1,0,1,2,0,1,2,0,3,1,2,0,3,1,4,2,0,3,1,4,2,0,5,3,1,4,2,0,5,3,1,6,4,2,
%T A025666 0,5,3,1,6,4,2,7,0,5,3,1,6,4,2,7,0,5,3,8,1,6,4,2,7,0,5,3,8,1,6,4,9,2,7,0,
%U A025666 5,3,8,1,6,4,9,2,7,0,5,10,3,8,1,6,4,9,2,7,0,5,10,3,8,1,6,11,4,9,2,7,0,5
%N A025666 Exponent of 7 (value of j) in n-th number of form 4^i*7^j.
%Y A025666 Adjacent sequences: A025663 A025664 A025665 this_sequence A025667 A025668 A025669
%Y A025666 Sequence in context: A113408 A022337 A025687 this_sequence A025679 A071491 A137298
%K A025666 nonn
%O A025666 1,6
%A A025666 David W. Wilson (davidwwilson(AT)comcast.net)

%I A025679
%S A025679 0,0,1,0,1,2,0,1,2,0,3,1,2,0,3,1,4,2,0,3,1,4,2,5,0,3,1,4,2,5,0,3,6,1,4,2,
%T A025679 5,0,3,6,1,4,7,2,5,0,3,6,1,4,7,2,5,8,0,3,6,1,4,7,2,5,8,0,3,6,9,1,4,7,2,5,
%U A025679 8,0,3,6,9,1,4,7,10,2,5,8,0,3,6,9,1,4,7,10,2,5,8,0,11,3,6,9,1,4,7,10,2,5
%N A025679 Exponent of 9 (value of j) in n-th number of form 5^i*9^j.
%Y A025679 Adjacent sequences: A025676 A025677 A025678 this_sequence A025680 A025681 A025682
%Y A025679 Sequence in context: A022337 A025687 A025666 this_sequence A071491 A137298 A136487
%K A025679 nonn
%O A025679 1,6
%A A025679 David W. Wilson (davidwwilson(AT)comcast.net)

%I A071491
%S A071491 1,2,0,1,2,0,3,1,2,3,1,2,4,3,5,0,3,4,2,6,1,2,4,1,3,0,2,1,7,2,4,7,8,
%T A071491 4,2,0,6,1,5,2
%N A071491 Sprague-Grundy values for octal game .334.
%D A071491 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982; see Chapter 4.
%Y A071491 Adjacent sequences: A071488 A071489 A071490 this_sequence A071492 A071493 A071494
%Y A071491 Sequence in context: A025687 A025666 A025679 this_sequence A137298 A136487 A021501
%K A071491 nonn
%O A071491 1,2
%A A071491 njas and Sue Pope (pope(AT)research.att.com), May 29 2002

%I A137298
%S A137298 1,0,1,1,0,1,0,2,0,1,2,0,4,0,1,0,6,0,6,0,1,6,0,18,0,9,0,1,0,30,0,42,0,
%T A137298 13,0,1,30,0,120,0,87,0,18,0,1,0,240,0,414,0,178,0,25,0,1,270,0,1320,0,
%U A137298 1197,0,340,0,34,0,1
%V A137298 1,0,1,-1,0,1,0,-2,0,1,2,0,-4,0,1,0,6,0,-6,0,1,-6,0,18,0,-9,0,1,0,-30,0,42,0,-13,0,1,
%W A137298 30,0,-120,0,87,0,-18,0,1,0,240,0,-414,0,178,0,-25,0,1,-270,0,1320,0,-1197,0,340,0,-34,
%X A137298 0,1
%N A137298 Triangle read by rows: cofficients of a Hermite-like set of recursive polynomials that appear by integration to be orthogonal using the substitution on the Hermite recursion of n->f(n) where f(n)=A000931[n] is the Padovan sequence.
%C A137298 The number-like behavior of the Padovan sequence made me think that I might get a orthogonal polynomial set by this substitution:
%C A137298 Table[Integrate[Exp[ -x2/2]*P[x,n]*P[x, n + 1], {x, -Infinity, Infinity}], {n, 0, 10}];
%C A137298 {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0};
%C A137298 The row sums are:
%C A137298 Table[Apply[Plus, CoefficientList[P[x, n], x]], {n, 0, 10}];
%C A137298 {1, 1, 0, -1, -1, 1, 4, 0, -20, -20, 160}
%C A137298 The tiling property of the Fibonacci and Padovan sequences makes me think that other sequence of fundamental number theory "beta Integer-like" sequences might give orthogonal polynomials as well.
%F A137298 a(n) = a(n-2)+a(n-3): A000931(n); p(x,0)=1;p(x,1)=x; p(x,n)=x*p(x,n-1)-a(n)*p(n,n-2)
%e A137298 {1},
%e A137298 {0, 1},
%e A137298 {-1, 0, 1},
%e A137298 {0, -2, 0, 1},
%e A137298 {2, 0, -4, 0, 1},
%e A137298 {0, 6, 0, -6, 0, 1},
%e A137298 {-6, 0, 18, 0, -9, 0, 1},
%e A137298 {0, -30, 0, 42, 0, -13, 0, 1},
%e A137298 {30, 0, -120, 0, 87, 0, -18, 0, 1},
%e A137298 {0, 240, 0, -414, 0, 178, 0, -25, 0, 1},
%e A137298 {-270, 0, 1320, 0, -1197, 0, 340, 0, -34, 0, 1}
%t A137298 f[0] = 0; f[1] = 1;f[2]=1; f[n_] := f[n] = f[n - 2] + f[n - 3]; P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = x*P[x, n - 1] - f[n]*P[x, n - 2]; Table[ExpandAll[P[x, n]], {n, 0, 10}]; a = Table[CoefficientList[P[x, n], x], {n, 0, 10}] Flatten[a]
%Y A137298 Cf. A000045, A000931.
%Y A137298 Adjacent sequences: A137295 A137296 A137297 this_sequence A137299 A137300 A137301
%Y A137298 Sequence in context: A025666 A025679 A071491 this_sequence A136487 A021501 A101662
%K A137298 nonn,uned,tabl
%O A137298 1,8
%A A137298 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 14 2008

%I A136487
%S A136487 1,1,1,1,1,1,1,1,0,2,0,1,2,0,4,0,2,3,2,7,4,5,2,1,5,5,11,11,7,7,1,1,8,12,
%T A136487 16,28,8,20,0,4,13,25,20,60,2,46,12,12,3,1,21,50,19,120,38,92,50,24,15,
%U A136487 2,1
%V A136487 1,1,1,1,1,-1,-1,-1,0,2,0,-1,2,0,-4,0,2,-3,2,7,-4,-5,2,1,5,-5,-11,11,7,-7,-1,1,-8,12,
%W A136487 16,-28,-8,20,0,-4,13,-25,-20,60,-2,-46,12,12,-3,-1,-21,50,19,-120,38,92,-50,-24,15,2,
%X A136487 -1
%N A136487 Coefficents of designed "null" term polynomial recursion: the cubic term is zero: p(x,n)=(x+1)*p(x,n-1)-(x^2-1)*p(x,n-2).
%C A136487 Row sums are:
%C A136487 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0}
%C A136487 A converse recursion is with different signs but same absolute coefficients is:
%C A136487 P[x, -1] = 0; P[x, 0] = 1; P[x, 1] = x - 1;
%C A136487 P[x_, n_] := P[x, n] = (x + 1)*P[x, n - 1] - (x^2 - 1)*P[x, n - 2]
%F A136487 p(x,-1)=0;p(x,0)=1;p(x,1)=x+1; p(x,n)=(x+1)*p(x,n-1)-(x^2-1)*p(x,n-2)
%e A136487 {1},
%e A136487 {1, 1},
%e A136487 {}, <- designed null tern
%e A136487 {1, 1, -1, -1},
%e A136487 {-1, 0, 2, 0, -1},
%e A136487 {2, 0, -4, 0, 2},
%e A136487 {-3, 2, 7, -4, -5, 2, 1},
%e A136487 {5, -5, -11, 11, 7, -7, -1, 1},
%e A136487 {-8, 12, 16, -28, -8, 20, 0, -4},
%e A136487 {13, -25, -20, 60, -2, -46, 12, 12, -3, -1},
%e A136487 {-21, 50, 19, -120, 38, 92, -50, -24, 15, 2, -1}
%t A136487 Clear[P,n,m,x] P[x,-1]=0;P[x,0]=1;P[x,1]=x-1; P[x_,n_]:=P[x,n]=(x+1)*P[x,n-1]-(x^2-1)*P[x,n-2]; a=Table[CoefficientList[P[x,n],x],{n,0,10}]; Flatten[a]
%Y A136487 Adjacent sequences: A136484 A136485 A136486 this_sequence A136488 A136489 A136490
%Y A136487 Sequence in context: A025679 A071491 A137298 this_sequence A021501 A101662 A091064
%K A136487 nonn,uned,tabf
%O A136487 1,10
%A A136487 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 21 2008

%I A021501
%S A021501 0,0,2,0,1,2,0,7,2,4,3,4,6,0,7,6,4,5,8,7,5,2,5,1,5,0,9,0,5,4,3,2,5,
%T A021501 9,5,5,7,3,4,4,0,6,4,3,8,6,3,1,7,9,0,7,4,4,4,6,6,8,0,0,8,0,4,8,2,8,
%U A021501 9,7,3,8,4,3,0,5,8,3,5,0,1,0,0,6,0,3,6,2,1,7,3,0,3,8,2,2,9,3,7,6,2
%N A021501 Decimal expansion of 1/497.
%Y A021501 Adjacent sequences: A021498 A021499 A021500 this_sequence A021502 A021503 A021504
%Y A021501 Sequence in context: A071491 A137298 A136487 this_sequence A101662 A091064 A075993
%K A021501 nonn,cons
%O A021501 0,3
%A A021501 njas

%I A101662
%S A101662 0,1,0,2,0,1,2,1,0,1,0,2,2,1,0,2,0,1,0,2,0,1,2,1,2,1,0,2,0,1,2,1,0,1,0,
%T A101662 2,0,1,2,1,0,1,0,2,2,1,0,2,2,1,0,2,0,1,2,1,0,1,0,2,2,1,0,2,0,1,0,2,0,1,
%U A101662 2,1,0,1,0,2,2,1,0,2,0,1,0,2,0,1,2,1,2,1,0,2,0,1,2,1,2,1,0,2,0,1,2,1,0
%N A101662 Fixed point of morphism 0 -> 01, 1 -> 02, 2 -> 21.
%F A101662 a(2n) = 2[a(n)==2], a(2n+1) = 1 + [a(n)==1].
%t A101662 Nest[ Function[ l, {Flatten[(l /. {0 -> {0, 1}, 1 -> {0, 2}, 2 -> {2, 1}})]}], {0}, 7] (from Robert G. Wilson v Feb 28 2005)
%Y A101662 Adjacent sequences: A101659 A101660 A101661 this_sequence A101663 A101664 A101665
%Y A101662 Sequence in context: A137298 A136487 A021501 this_sequence A091064 A075993 A117170
%K A101662 nonn,easy
%O A101662 0,4
%A A101662 Ralf Stephan, Dec 11 2004

%I A091064
%S A091064 0,0,0,1,0,1,0,2,0,1,2,1,0,1,0,3,0,1,2,1,0,3,2,1,0,1,0,2,0,1,0,4,0,1,2,
%T A091064 1,3,1,2,1,0,1,4,1,0,3,2,1,0,1,0,2,0,1,3,2,0,1,0,2,0,1,0,5,0,1,2,1,3,1,
%U A091064 2,1,0,4,2,1,3,1,2,1,0,1,2,1,0,5,2,1,0,1,2,4,0,3,2,1,0,1,0,2,0,1
%N A091064 In binary representation: length of longest proper prefix of n, that is also a suffix.
%C A091064 a(n) <= A070939(n)-1 by definition;
%C A091064 a(2^k-1)=k-1; for k>0: a(2^k+1)=1; for k>2: a(2^k+2)=2;
%C A091064 a(A091065(n)) = 0, a(A091066(n)) > 0.
%H A091064 <a href="http://www.research.att.com/~njas/sequences/Sindx_Bi.html#binary">Index entries for sequences related to binary expansion of n</a>
%Y A091064 Cf. A007088.
%Y A091064 Adjacent sequences: A091061 A091062 A091063 this_sequence A091065 A091066 A091067
%Y A091064 Sequence in context: A136487 A021501 A101662 this_sequence A075993 A117170 A117466
%K A091064 nonn,base
%O A091064 0,8
%A A091064 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Dec 17 2003

%I A075993
%S A075993 1,1,1,2,0,1,2,1,0,1,3,1,0,0,1,3,1,1,0,0,1,4,1,1,0,0,0,1,4,2,0,1,0,0,0,
%T A075993 1,5,1,1,1,0,0,0,0,1,5,2,1,0,1,0,0,0,0,1,6,2,1,0,1,0,0,0,0,0,1,6,2,1,1,
%U A075993 0,1,0,0,0,0,0,1,7,2,1,1,0,1,0,0,0,0,0,0,1,7,3,1,1,0,0,1,0,0,0,0,0,0,1
%N A075993 Triangular array T(n,m) = number of integers k such that Floor(n/k)=m. Row n has n terms, for n=1,2,3,...
%C A075993 The sum of numbers in row n is n.
%C A075993 Number of terms > 0 per row: Sum(A063524(T(n,k)):1<=k<=n} = A055086(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 06 2006
%F A075993 T(n, m)=Floor(n/m)-Floor(n/(m+1)).
%e A075993 T(5, 1) = 3 counts k such that Floor(5/k) = 1, namely k = 5, 4, 3. First 6 rows:
%e A075993 1
%e A075993 1 1
%e A075993 2 0 1
%e A075993 2 1 0 1
%e A075993 3 1 0 0 1
%e A075993 3 1 1 0 0 1
%Y A075993 Columns 1, 2, 3 are essentially A004526, A008615, A008679.
%Y A075993 Adjacent sequences: A075990 A075991 A075992 this_sequence A075994 A075995 A075996
%Y A075993 Sequence in context: A021501 A101662 A091064 this_sequence A117170 A117466 A136266
%K A075993 nonn,tabl
%O A075993 1,4
%A A075993 Clark Kimberling (ck6(AT)evansville.edu), Sep 28 2002

%I A117170
%S A117170 1,1,1,2,0,1,2,1,0,1,3,1,0,0,1,3,2,1,0,0,1,4,1,1,0,0,0,1,3,3,1,1,0,0,0,
%T A117170 1,6,1,2,1,0,0,0,0,1,5,4,1,1,1,0,0,0,0,1,5,2,2,1,1,0,0,0,0,0,1,6,4,2,2,
%U A117170 1,1,0,0,0,0,0,1,7,2,2,1,1,1,0,0,0,0,0,0,1,7,6,2,2,1,1,1,0,0,0,0,0,0,1
%N A117170 Triangle of coefficients for the Inverse Shift-Moebius transform, read by rows.
%C A117170 Column k = Inverse-Shift-Moebius transform of all zeros except
%C A117170 for a single '1' in position k: [0,0,0,..(k-1)zeros..,1,0,0,0,...].
%C A117170 Column 1 is A117171, and equals Inverse-Shift-Moebius([1,0,0,0,...]).
%C A117170 Column 2 is A117172, and equals Inverse-Shift-Moebius([0,1,0,0,...]).
%C A117170 Column 3 is A117173, and equals Inverse-Shift-Moebius([0,0,1,0,...]).
%C A117170 Row sums give A117174, and equals Inverse-Shift-Moebius([1,1,1,...]).
%e A117170 Triangle begins:
%e A117170 1;
%e A117170 1, 1;
%e A117170 2, 0, 1;
%e A117170 2, 1, 0, 1;
%e A117170 3, 1, 0, 0, 1;
%e A117170 3, 2, 1, 0, 0, 1;
%e A117170 4, 1, 1, 0, 0, 0, 1;
%e A117170 3, 3, 1, 1, 0, 0, 0, 1;
%e A117170 6, 1, 2, 1, 0, 0, 0, 0, 1;
%e A117170 5, 4, 1, 1, 1, 0, 0, 0, 0, 1;
%e A117170 5, 2, 2, 1, 1, 0, 0, 0, 0, 0, 1;
%e A117170 6, 4, 2, 2, 1, 1, 0, 0, 0, 0, 0, 1;
%e A117170 7, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1;
%e A117170 7, 6, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1;
%e A117170 10, 3, 4, 1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1;
%e A117170 7, 6, 2, 3, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1; ...
%o A117170 (PARI) {T(n,k)=if(n<k,0,prod(i=0,n, matrix(n,n,r,c,if(r>=c,if((r+i)%(c+i)==0,1,0))))[n,k])}
%Y A117170 Cf. A117171 (column 1), A117172 (column 2), A117173 (column 3), A117174 (row sums), A117165 (inverse), A117162, A008683; A117176.
%Y A117170 Adjacent sequences: A117167 A117168 A117169 this_sequence A117171 A117172 A117173
%Y A117170 Sequence in context: A101662 A091064 A075993 this_sequence A117466 A136266 A054523
%K A117170 nonn,tabl
%O A117170 1,4
%A A117170 Wouter Meeussen (wouter.meeussen(AT)pandora.be) and Paul D. Hanna (pauldhanna(AT)juno.com), Mar 05 2006

%I A117466
%S A117466 1,1,1,2,0,1,2,1,0,1,3,1,0,0,1,4,1,1,0,0,1,5,1,1,0,0,0,1,6,2,0,1,0,0,0,
%T A117466 1,8,2,1,1,0,0,0,0,1,10,2,1,0,1,0,0,0,0,1,12,3,1,0,1,0,0,0,0,0,1,15,3,2,
%U A117466 1,0,1,0,0,0,0,0,1,18,4,1,1,0,1,0,0,0,0,0,0,1,22,5,1,1,0,0,1,0,0,0,0,0
%N A117466 Triangle read by rows: T(n,k) is the number of partitions of n in which every integer from the smallest part k to the largest part occurs (1<=k<=n).
%C A117466 Row sums yield A034296. T(n,1)=A000009(n). sum(k*T(n,k),k=1..n)=A117467(n).
%F A117466 G.f.=G(t,x)=sum(tx^j*product(1+x^i, i=1..j-1)/(1-tx^j), j=1..infinity).
%e A117466 T(11,2)=3 because we have [4,3,2,2],[3,3,3,2], and [3,2,2,2,2].
%e A117466 Triangle starts:
%e A117466 1;
%e A117466 1,1;
%e A117466 2,0,1;
%e A117466 2,1,0,1;
%e A117466 3,1,0,0,1;
%e A117466 4,1,1,0,0,1;
%p A117466 g:=sum(t*x^j*product(1+x^i,i=1..j-1)/(1-t*x^j),j=1..50): gser:=simplify(series(g,x=0,17)): for n from 1 to 14 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 14 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form
%Y A117466 Cf. A034296, A000009, A117467.
%Y A117466 Adjacent sequences: A117463 A117464 A117465 this_sequence A117467 A117468 A117469
%Y A117466 Sequence in context: A091064 A075993 A117170 this_sequence A136266 A054523 A106351
%K A117466 nonn,tabl
%O A117466 1,4
%A A117466 Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 19 2006

%I A136266
%S A136266 1,0,1,0,1,1,0,1,2,0,1,2,1,0,1,3,1,0,1,3,2,0,1,3,2,1,0,1,3,3,1,0,1,4,3,
%T A136266 1,0,1,4,4,1,0,1,4,4,2,0,1,4,4,2,1,0,1,4,5,2,1,0,1,4,5,3,1,0,1,4,5,4,1,
%U A136266 0,1,4,6,4,1,0,1,5,6,4,1,0,1,5,7,4,1,0,1,5,7,5,1,0,1,5,7,5,2
%N A136266 Kappraff polynomials: Coefficients of a new type of recursive polynomial based on Conway's A04001 chaotic sequence:B(x, n) = x*B(x, A004001(n - 1)) + B(x, n - A004001(n - 1)): (I'm naming these Kappraff polynomials for the inspiration that Dr. Jay Kappraff has had in the area of sequence research over the years. Without him and Gary Adamson, nothing would have been done by me in this area. ).
%C A136266 Row sums are:
%C A136266 {1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}
%C A136266 This sequence comes from trying to get a chaotic sequence like behavior in
%C A136266 a recursive polynomial. As far as I know this is a net type of triangular sequence
%C A136266 which I based on the similarity of the A049310
%C A136266 B(x,n)=x*B(x,n-1)+B(x,n-2)
%C A136266 to the Fibonacci sequence.
%C A136266 I use the Conway A004001 chaotic sequence to index this such that
%C A136266 I substitute:
%C A136266 n-1-> Conway[n - 1]
%C A136266 n-2-> n - Conway[n - 1]
%D A136266 http://www-ec.njit.edu/~kappraff/
%F A136266 B(x, n) = x*B(x, A004001(n - 1)) + B(x, n - A004001(n - 1))
%e A136266 {1},
%e A136266 {0, 1},
%e A136266 {0, 1, 1},
%e A136266 {0, 1, 2},
%e A136266 {0, 1, 2, 1},
%e A136266 {0, 1, 3, 1},
%e A136266 {0, 1, 3, 2},
%e A136266 {0, 1, 3, 2, 1},
%e A136266 {0, 1, 3, 3, 1},
%e A136266 {0, 1, 4, 3, 1},
%e A136266 {0, 1, 4, 4, 1},
%e A136266 {0, 1, 4, 4, 2},
%e A136266 {0, 1, 4, 4, 2, 1},
%e A136266 {0, 1, 4, 5, 2, 1},
%e A136266 {0, 1, 4, 5, 3, 1},
%e A136266 {0, 1, 4, 5, 4, 1},
%e A136266 {0, 1, 4, 6, 4, 1},
%e A136266 {0, 1, 5, 6, 4, 1},
%e A136266 {0, 1, 5, 7, 4, 1},
%e A136266 {0, 1, 5, 7, 5, 1},
%e A136266 {0, 1, 5, 7, 5, 2}
%t A136266 Clear[Conway] Conway[0] = 1; Conway[1] = 1; Conway[2] = 1; Conway[n_] := Conway[n] = Conway[Conway[n - 1]] + Conway[n - Conway[n - 1]]; Clear[B, x, n]; B[x, 0] = 1; B[x, 1] = x; B[x_, n_] := B[x, n] = x*B[x, Conway[n - 1]] + B[x, n - Conway[n - 1]]; Table[ExpandAll[B[x, n]], {n, 0, 10}]; a = Table[CoefficientList[B[x, n], x], {n, 0, 10}] Flatten[a]
%Y A136266 Cf. A004001, A049310.
%Y A136266 Adjacent sequences: A136263 A136264 A136265 this_sequence A136267 A136268 A136269
%Y A136266 Sequence in context: A075993 A117170 A117466 this_sequence A054523 A106351 A096800
%K A136266 nonn,uned,tabf
%O A136266 1,9
%A A136266 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 18 2008

%I A054523
%S A054523 1,1,1,2,0,1,2,1,0,1,4,0,0,0,1,2,2,1,0,0,1,6,0,0,0,0,0,1,4,2,0,
%T A054523 1,0,0,0,1,6,0,2,0,0,0,0,0,1,4,4,0,0,1,0,0,0,0,1,10,0,0,0,0,0,0,
%U A054523 0,0,0,1,4,2,2,2,0,1,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,1,6,6
%N A054523 Triangle read by rows: T(n,k) = phi(n/k) if k divides n, T(n,k)=0 otherwise (n >= 1, 1<=k<=n).
%C A054523 Comments from Gary Adamson, Jan 08 2007: (Start) Let H be this lower triangular matrix. Then:
%C A054523 H * A051731 = A126988,
%C A054523 H * [1, 2, 3,...] = 1, 3, 5, 8, 9, 15,...= A018804,
%C A054523 H * sigma(n) = A038040 = d(n) * n = 1, 4, 6, 12, 10,... where sigma(n) = A000203,
%C A054523 H * d(n) (A000005) = sigma(n) = A000203,
%C A054523 Row sums of H = A018804 = sum of GCD (k,n),
%C A054523 H^2 * d(n) = d(n)*n, H^2 = A127192,
%C A054523 H * mu(n) (A008683) = phi(n) = A000010,
%C A054523 H^2 row sums = A018804. (End)
%Y A054523 Cf. A054521.
%Y A054523 Adjacent sequences: A054520 A054521 A054522 this_sequence A054524 A054525 A054526
%Y A054523 Sequence in context: A117170 A117466 A136266 this_sequence A106351 A096800 A036586
%K A054523 nonn,tabl
%O A054523 1,4
%A A054523 njas, Apr 09 2000

%I A106351
%S A106351 1,1,0,1,2,0,1,2,1,0,1,4,2,0,0,1,4,7,2,0,0,1,6,9,6,1,0,0,1,6,15,14,3,0,
%T A106351 0,0,1,8,21,24,15,2,0,0,0,1,8,28,46,30,10,1,0,0,0,1,10,35,66,68,30,4,0,
%U A106351 0,0,0,1,10,46,100,119,76,24,2,0,0,0,0,1,12,54,138,204,168,69,14,1,0,0
%N A106351 Triangle read by rows: T(n,k) = number of compositions of n into k parts such that no two adjacent parts are equal.
%H A106351 A. Knopfmacher and H. Prodinger, <a href="http://www.wits.ac.za/science/number_theory/apublic.htm">On Carlitz compositions</a>, European Journal of Combinatorics, Vol. 19, 1998, pp. 579-589.
%F A106351 G.f. 1/(1 - sum(k>0, (-1)^(k+1)*x^k*y^k/(1-x^k)).
%e A106351 1; 1,0; 1,2,0; 1,2,1,0; 1,4,2,0,0; ...
%e A106351 T(6,3)=7 because the compositions of 6 into 3 parts with no adjacent equal parts are 3+2+1, 3+1+2, 2+3+1, 2+1+3, 1+3+2, 1+2+3, 1+4+1.
%Y A106351 Row sums: A003242. Columns 3-6: A106352-A106355.
%Y A106351 Adjacent sequences: A106348 A106349 A106350 this_sequence A106352 A106353 A106354
%Y A106351 Sequence in context: A117466 A136266 A054523 this_sequence A096800 A036586 A092928
%K A106351 nonn,tabl
%O A106351 1,5
%A A106351 Christian G. Bower (bowerc(AT)usa.net), Apr 29 2005

%I A096800
%S A096800 1,1,1,2,0,1,2,1,0,1,4,5,5,0,1,2,2,5,6,0,1,6,28,28,7,7,0,1,4,90,136,49,8,8,0,1,6,
%T A096800 738,1082,432,90,9,9,0,1,4,6279,9525,4075,969,145,10,10,0,1,10,66594,101915,
%U A096800 44803,11143,1881,220,11,11,0,1,4,816362,1260268,565988,144300,25207,3300,318
%V A096800 1,1,1,2,0,1,2,1,0,1,4,-5,5,0,1,2,2,-5,6,0,1,6,-28,28,-7,7,0,1,4,90,-136,49,-8,8,0,1,6,
%W A096800 -738,1082,-432,90,-9,9,0,1,4,6279,-9525,4075,-969,145,-10,10,0,1,10,-66594,101915,
%X A096800 -44803,11143,-1881,220,-11,11,0,1,4,816362,-1260268,565988,-144300,25207,-3300,318
%N A096800 Triangle of coefficients, read by row polynomials P_n(y), that satisfy the g.f.: A096651(x,y) = Product_{n>=1} 1/(1-x^n)^[P_n(y)/n], with P_n(0)=0 for n>=1.
%C A096800 Row sums form the positive integers. The first column forms the totients (A000010). The inverse Moebius transform of each column forms the columns of triangle {n/k*A096799(n,k)}. A generalized Euler transform of the row polynomials of this triangle generates A096651; the row sums of A096651^n form the n-dimensional partitions.
%e A096800 G.f.: 1/A096651(x,y) = (1-x)^y*(1-x^2)^[(y+y^2)/2]*(1-x^3)^[(2y+y^3)/3]*(1-x^4)^[(2y+y^2+y^4)/4]*(1-x^5)^[(4y-5y^2+5y^3+y^5)/5]*...
%e A096800 Rows begin:
%e A096800 [1],
%e A096800 [1,1],
%e A096800 [2,0,1],
%e A096800 [2,1,0,1],
%e A096800 [4,-5,5,0,1],
%e A096800 [2,2,-5,6,0,1],
%e A096800 [6,-28,28,-7,7,0,1],
%e A096800 [4,90,-136,49,-8,8,0,1],
%e A096800 [6,-738,1082,-432,90,-9,9,0,1],
%e A096800 [4,6279,-9525,4075,-969,145,-10,10,0,1],
%e A096800 [10,-66594,101915,-44803,11143,-1881,220,-11,11,0,1],
%e A096800 [4,816362,-1260268,565988,-144300,25207,-3300,318,-12,12,0,1],
%e A096800 [12,-11418459,17738565,-8095100,2105129,-375609,50414,-5382,442,-13,13,0,1],...
%Y A096800 Cf. A096651, A096799.
%Y A096800 Adjacent sequences: A096797 A096798 A096799 this_sequence A096801 A096802 A096803
%Y A096800 Sequence in context: A136266 A054523 A106351 this_sequence A036586 A092928 A085097
%K A096800 sign,tabl
%O A096800 0,4
%A A096800 Paul D. Hanna (pauldhanna(AT)juno.com), Jul 13 2004

%I A036586
%S A036586 2,0,1,2,1,0,2,0,1,0,2,1,2,0,1,2,1,0,2,1,2,0,1,0,2,0,1,2,1,0,2,0,1,
%T A036586 0,2,1,2,0,1,0,2,0,1,2,1,0,2,1,2,0,1,2,1,0,2,0,1,0,2,1,2,0,1,2,1,0,
%U A036586 2,1,2,0,1,0,2,0,1,2,1,0,2,1,2,0,1,2,1,0,2,0,1,0,2,1,2,0,1,0,2,0,1
%N A036586 Ternary Thue-Morse sequence: closed under a->abc, b->ac, c->b.
%D A036586 M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 26.
%Y A036586 Adjacent sequences: A036583 A036584 A036585 this_sequence A036587 A036588 A036589
%Y A036586 Sequence in context: A054523 A106351 A096800 this_sequence A092928 A085097 A117997
%K A036586 nonn
%O A036586 0,1
%A A036586 njas

%I A092928
%S A092928 0,0,1,0,2,0,1,2,1,0,2,0,1,7,1,0,12,0,1,2,1,0,2,5,1,2,2,0,7,0,5,2,12,1,
%T A092928 8,0,4,5,1,0,22,0,2,4,14,0,9,7,5,5,1,0,2,4,1,13,6,0,25,0,5,9,1,5,4,0,1,
%U A092928 4,8,0,10,0,5,8,5,1,4,0,1,12,1,0,7,5,2,2,7,0,12,1,1,8,2,4,5,0,8,7,14,0
%V A092928 0,0,1,0,2,0,1,2,1,0,2,0,1,7,1,0,12,0,-1,2,1,0,2,5,1,2,2,0,7,0,5,2,12,1,8,0,4,5,1,0,22,
%W A092928 0,2,4,14,0,9,7,5,5,1,0,2,4,1,13,6,0,25,0,5,9,1,5,4,0,1,4,8,0,10,0,5,8,5,1,4,0,1,12,1,
%X A092928 0,7,5,2,2,7,0,12,1,1,8,2,4,5,0,8,7,14,0
%N A092928 Value of k pertaining to A092927: value of k such that n(n-1)(n-2)...(n-k)+1 is the least prime of this form; a(n)=-1 if no such k exists, i.e. A092927(n)=0.
%C A092928 Subsidiary sequence: First occurrence of n in this sequence. 1,3,5,...
%e A092928 a(8) = 2 as 8*7*6 +1 = 337 is the least prime of this form.
%o A092928 (PARI) f(n,k)=prod(i=n-k,n,i)+1; a(n)=for (i=0,n,if(isprime(f(n,i)),print1(i); break()))
%Y A092928 Cf. A092925, A092926, A092927.
%Y A092928 Adjacent sequences: A092925 A092926 A092927 this_sequence A092929 A092930 A092931
%Y A092928 Sequence in context: A106351 A096800 A036586 this_sequence A085097 A117997 A079684
%K A092928 sign
%O A092928 1,5
%A A092928 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 18 2004
%E A092928 More terms from Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 19 2004
%E A092928 Corrected and extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Mar 23 2004

%I A085097
%S A085097 1,1,2,0,1,2,1,0,3,1,1,0,1,1,2,0,1,3,1,0,2,1,1,0,0,1,0,0,1,2,1,0,2,1,1,
%T A085097 0,1,1,2,0,1,2,1,0,3,1,1,0,0,0,2,0,1,0,1,0,2,1,1,0,1,1,3,0,1,2,1,0,2,1,1,
%U A085097 0,1,1,0,0,1,2,1,0,0,1,1,0,1,1,2,0,1,3,1,0,2,1,1,0,1,0,3,0,1,2,1,0,2,1,1,0
%V A085097 1,-1,2,0,-1,-2,-1,0,-3,1,-1,0,-1,1,-2,0,-1,3,-1,0,-2,1,-1,0,0,1,0,0,-1,2,-1,0,-2,1,1,
%W A085097 0,-1,1,-2,0,-1,2,-1,0,3,1,-1,0,0,0,-2,0,-1,0,1,0,-2,1,-1,0,-1,1,3,0,1,2,-1,0,-2,-1,-1,
%X A085097 0,-1,1,0,0,1,2,-1,0,0,1,-1,0,1,1,-2,0,-1,-3,1,0,-2,1,1,0,-1,0,3,0,-1,2,-1,0,2,1,-1,0
%N A085097 Ramanujan sum c_3(n).
%D A085097 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
%F A085097 a(n) = phi(n)*mu(n/gcd(n, 3)) / phi(n/gcd(n, 3)).
%o A085097 (PARI) a(n)=eulerphi(n)*moebius(n/gcd(n,3))/eulerphi(n/gcd(n,3))
%Y A085097 Cf. A086831.
%Y A085097 Adjacent sequences: A085094 A085095 A085096 this_sequence A085098 A085099 A085100
%Y A085097 Sequence in context: A096800 A036586 A092928 this_sequence A117997 A079684 A033761
%K A085097 sign,easy,mult
%O A085097 1,3
%A A085097 Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 10 2003
%E A085097 More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 12 2003

%I A117997
%S A117997 1,1,2,0,1,2,1,0,6,1,1,0,1,1,2,0,1,6,1,0,2,1,1,0,0,1,18,0,1,2,1,0,2,1,1,
%T A117997 0,1,1,2,0,1,2,1,0,6,1,1,0,0,0,2,0,1,18,1,0,2,1,1,0,1,1,6,0,1,2,1,0,2,1,
%U A117997 1,0,1,1,0,0,1,2,1,0,54,1,1,0,1,1,2,0,1,6,1,0,2,1,1,0,1,0,6,0,1,2,1,0,2
%V A117997 1,-1,2,0,-1,-2,-1,0,6,1,-1,0,-1,1,-2,0,-1,-6,-1,0,-2,1,-1,0,0,1,18,0,-1,2,-1,0,-2,1,1,
%W A117997 0,-1,1,-2,0,-1,2,-1,0,-6,1,-1,0,0,0,-2,0,-1,-18,1,0,-2,1,-1,0,-1,1,-6,0,1,2,-1,0,-2,
%X A117997 -1,-1,0,-1,1,0,0,1,2,-1,0,54,1,-1,0,1,1,-2,0,-1,6,1,0,-2,1,1,0,-1,0,-6,0,-1,2,-1,0,2
%N A117997 Sum_{d|n} a(d) = n for n=3^m (m>=0) and for other n the sum is zero; i.e., the Moebius transform of [1,0,3,0,0,0,0,0,9,0,...].
%F A117997 G.f.: x = Sum_{n>=1} a(n)/n*log(1 + x^n + x^(2n)).
%o A117997 (PARI) {a(n)=if(n==1,1,-n*polcoeff(x+sum(k=1,n-1,a(k)/k*subst(log(1+x+x^2+x*O(x^n)),x,x^k+x*O(x^n))),n))}
%Y A117997 Adjacent sequences: A117994 A117995 A117996 this_sequence A117998 A117999 A118000
%Y A117997 Sequence in context: A036586 A092928 A085097 this_sequence A079684 A033761 A033805
%K A117997 sign
%O A117997 0,3
%A A117997 Paul D. Hanna (pauldhanna(AT)juno.com), Apr 08 2006

%I A079684
%S A079684 0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,2,0,1,2,1,1,0,0,1,5,3,2,1,3,3,2,1,7,2,3,
%T A079684 7,4,5,1,5,3,5,3,9,3,5,1,5,7,6,6,6,4,9,8,5,3,4,5,8,8,4,8,5,9,7,7,6,9,10,
%U A079684 5,7,8,6,10,7,11,7,9,10,8,8,15,10,13,8,10,13,8,12,10,6,18,12,12,15,9,12
%N A079684 Number of 3's in n!.
%H A079684 <a href="http://www.research.att.com/~njas/sequences/Sindx_Fa.html#factorial">Index entries for sequences related to factorial numbers</a>.
%F A079684 a(n) = A034886(n) - (A027869(n) + A079680(n) + A079714(n) + A079688(n) + A079690(n) + A079691(n) + A079692(n) + A079693(n) + A079694(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 27 2008
%Y A079684 Cf. A079714.
%Y A079684 Cf. A000142, A137579, A137580.
%Y A079684 Adjacent sequences: A079681 A079682 A079683 this_sequence A079685 A079686 A079687
%Y A079684 Sequence in context: A092928 A085097 A117997 this_sequence A033761 A033805 A033797
%K A079684 easy,nonn
%O A079684 0,16
%A A079684 Cino Hilliard (hillcino368(AT)gmail.com), Jan 31 2003
%E A079684 Corrected by Jason Earls (jcearls(AT)cableone.net), Jul 06 2003

%I A033761
%S A033761 1,1,1,2,0,1,2,1,1,1,1,0,3,1,0,2,1,1,1,0,1,3,1,2,0,0,1,2,1,0,3,1,0,2,1,
%T A033761 1,2,0,1,0,2,1,2,1,0,3,0,1,3,0,0,2,1,0,0,1,2,4,1,1,0,1,1,1,0,1,3,1,1,0,
%U A033761 1,1,2,1,0,3,0,1,4,0,1,0,1,0,2,1,1,2,0,0,2,2,1,3,0,0,2,2,1,0,2,1,0,1,0
%N A033761 Product t2(q^d); d | 2, where t2 = theta2(q)/(2*q^(1/4)).
%C A033761 Also the number of representations of n as the sum of a triangular number and twice a triangular number. - James A. Sellers (sellersj(AT)math.psu.edu), Dec 21 2005
%D A033761 M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
%F A033761 Euler transform of period 4 sequence [1, 0, 1, -2, ...]. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 14 2004
%F A033761 Expansion of q^(-3/8)eta(q^2)eta^2(q^4)/eta(q) in powers of q. - Michael Somos Jul 05 2006
%F A033761 Expansion of q^(-3/4)(theta_2(q)theta_2(q^2))/4 in powers of q^2. - Michael Somos Jul 05 2006
%F A033761 Expansion of psi(q)psi(q^2) in powers of q where psi() is a Ramanujan theta function.
%F A033761 Given g.f. A(x), then B(x)=x^3*A(x^8) satisfies 0=f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6) = +u1^4*u6^2 +3*u2^2*u3^4 -4*u1*u2*u3*u6*(u2^2 +3*u6^2) - Michael Somos Jul 05 2006
%o A033761 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^4+A)^2/eta(x+A), n))} /* Michael Somos Jul 05 2006 */
%Y A033761 Cf. A097723.
%Y A033761 Adjacent sequences: A033758 A033759 A033760 this_sequence A033762 A033763 A033764
%Y A033761 Sequence in context: A085097 A117997 A079684 this_sequence A033805 A033797 A033793
%K A033761 nonn
%O A033761 0,4
%A A033761 njas
%E A033761 More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 14 2004

%I A033805
%S A033805 1,1,1,2,0,1,2,1,1,1,1,0,3,1,0,2,1,1,1,0,1,3,1,3,1,1,3,
%T A033805 2,2,2,4,2,1,3,1,4,3,0,3,1,3,2,2,2,3,4,3,2,4,3,2
%N A033805 Product t2(q^d); d | 46, where t2 = theta2(q)/(2*q^(1/4)).
%Y A033805 Adjacent sequences: A033802 A033803 A033804 this_sequence A033806 A033807 A033808
%Y A033805 Sequence in context: A117997 A079684 A033761 this_sequence A033797 A033793 A033785
%K A033805 nonn
%O A033805 0,4
%A A033805 njas

%I A033797
%S A033797 1,1,1,2,0,1,2,1,1,1,1,0,3,1,0,2,1,1,1,1,2,4,3,2,1,2,2,
%T A033797 3,2,1,3,4,1,2,3,2,3,1,2,2,6,4,4,2,2,5,3,3,4,3,4
%N A033797 Product t2(q^d); d | 38, where t2 = theta2(q)/(2*q^(1/4)).
%Y A033797 Adjacent sequences: A033794 A033795 A033796 this_sequence A033798 A033799 A033800
%Y A033797 Sequence in context: A079684 A033761 A033805 this_sequence A033793 A033785 A033781
%K A033797 nonn
%O A033797 0,4
%A A033797 njas

%I A033793
%S A033793 1,1,1,2,0,1,2,1,1,1,1,0,3,1,0,2,1,2,2,1,3,3,2,4,1,1,2,
%T A033793 3,1,3,4,1,2,3,3,3,3,3,4,2,6,2,3,3,3,4,3,5,4,2,3
%N A033793 Product t2(q^d); d | 34, where t2 = theta2(q)/(2*q^(1/4)).
%Y A033793 Adjacent sequences: A033790 A033791 A033792 this_sequence A033794 A033795 A033796
%Y A033793 Sequence in context: A033761 A033805 A033797 this_sequence A033785 A033781 A127242
%K A033793 nonn
%O A033793 0,4
%A A033793 njas

%I A033785
%S A033785 1,1,1,2,0,1,2,1,1,1,1,0,3,2,1,3,3,1,2,2,2,4,2,3,0,3,3,
%T A033785 3,4,3,4,3,2,4,5,3,5,0,4,4,6,6,7,5,4,7,5,7,7,6,0
%N A033785 Product t2(q^d); d | 26, where t2 = theta2(q)/(2*q^(1/4)).
%Y A033785 Adjacent sequences: A033782 A033783 A033784 this_sequence A033786 A033787 A033788
%Y A033785 Sequence in context: A033805 A033797 A033793 this_sequence A033781 A127242 A025853
%K A033785 nonn
%O A033785 0,4
%A A033785 njas

%I A033781
%S A033781 1,1,1,2,0,1,2,1,1,1,1,1,4,2,2,2,2,3,2,1,2,4,2,6,2,2,3,
%T A033781 4,4,2,4,3,4,5,8,4,6,3,6,6,5,6,6,6,3,12,3,3,8,5,3
%N A033781 Product t2(q^d); d | 22, where t2 = theta2(q)/(2*q^(1/4)).
%Y A033781 Adjacent sequences: A033778 A033779 A033780 this_sequence A033782 A033783 A033784
%Y A033781 Sequence in context: A033797 A033793 A033785 this_sequence A127242 A025853 A025847
%K A033781 nonn
%O A033781 0,4
%A A033781 njas

%I A127242
%S A127242 1,0,1,1,1,1,2,0,1,2,1,1,2,0,1,2,2,2,4,2,3,3,2,2,4,0,1,3,1,2,3,3,3,4,3,
%T A127242 2,5,2,5,5,5,2,4,5,2,7,3,6,5,1,7,7,3,4,5,6,4,4,4,2,9
%N A127242 Diagonal sums of Thue-Morse triangle A127241.
%F A127242 a(n)=sum{k=0..floor(n/2), A010060(binomial(k,n-2k))}
%Y A127242 Adjacent sequences: A127239 A127240 A127241 this_sequence A127243 A127244 A127245
%Y A127242 Sequence in context: A033793 A033785 A033781 this_sequence A025853 A025847 A092130
%K A127242 easy,nonn
%O A127242 0,7
%A A127242 Paul Barry (pbarry(AT)wit.ie), Jan 10 2007

%I A025853
%S A025853 1,0,0,1,0,0,1,0,1,1,0,1,2,0,1,2,1,1,2,1,2,2,1,2,4,1,2,
%T A025853 4,2,2,4,2,4,4,2,4,6,2,4,6,4,4,6,4,6,6,4,6,9,4,6,9,6,6,
%U A025853 9,6,9,9,6,9,12,6,9,12,9,9,12,9,12,12,9,12,16,9,12,16,12
%N A025853 Expansion of 1/((1-x^3)(1-x^8)(1-x^12)).
%Y A025853 Adjacent sequences: A025850 A025851 A025852 this_sequence A025854 A025855 A025856
%Y A025853 Sequence in context: A033785 A033781 A127242 this_sequence A025847 A092130 A029298
%K A025853 nonn
%O A025853 0,13
%A A025853 njas

%I A025847
%S A025847 1,0,0,1,0,0,1,1,0,1,2,0,1,2,1,1,2,2,1,2,3,2,2,3,3,2,3,
%T A025847 4,3,3,5,4,3,5,5,4,5,6,5,5,7,6,6,7,7,7,7,8,8,8,9,9,9,9,
%U A025847 10,10,10,11,11,11,12,12,12,13,13,13,14,14,14,15,16,15
%N A025847 Expansion of 1/((1-x^3)(1-x^7)(1-x^10)).
%Y A025847 Adjacent sequences: A025844 A025845 A025846 this_sequence A025848 A025849 A025850
%Y A025847 Sequence in context: A033781 A127242 A025853 this_sequence A092130 A029298 A059835
%K A025847 nonn
%O A025847 0,11
%A A025847 njas

%I A092130
%S A092130 1,0,0,0,1,0,0,1,0,0,1,1,0,1,1,0,1,2,0,1,2,1,1,3,1,1,3,2,1,4,3,1,4,4,2,
%T A092130 5,5,2,5,7,3,6,8,4,6,10,6,7,12,7,8
%N A092130 Number of partitions of n into distinct parts == 1 mod 3, with 1 as the smallest part.
%C A092130 Also number of partitions of n such that if k is the largest part, then k occurs exactly once and integers from 1 to k-1 occur a positive multiple of 3 times. Example: a(18)=2 because we have [3,2,2,2,1,1,1,1,1,1,1,1,1] and [3,2,2,2,2,2,2,1,1,1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 18 2006
%F A092130 G.f.=x*product(1+x^(1+3k), k=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 18 2006
%e A092130 For a(24), we have 19+4+1, 16+7+1, 13+10+1, so a(24)=3.
%p A092130 g:=x*product(1+x^(1+3*k),k=1..25): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=1..51); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 18 2006
%o A092130 (PARI) for(i=0,50,print1(","polcoeff(prod(k=1,50,(1+x^(3*k+1))),i)))
%Y A092130 Cf. A027349.
%Y A092130 Adjacent sequences: A092127 A092128 A092129 this_sequence A092131 A092132 A092133
%Y A092130 Sequence in context: A127242 A025853 A025847 this_sequence A029298 A059835 A093998
%K A092130 nonn
%O A092130 1,18
%A A092130 Jon Perry (perry(AT)globalnet.co.uk), Mar 30 2004

%I A029298
%S A029298 1,0,0,1,0,0,2,0,1,2,1,1,3,1,2,3,3,2,5,3,4,5,5,4,8,5,7,
%T A029298 8,8,7,12,8,11,12,12,11,17,12,16,17,18,16,23,18,22,23,25,
%U A029298 22,31,25,30,31,33,30,40,33,40,40,43,40,51,43,51,51,55
%N A029298 Expansion of 1/((1-x^3)(1-x^6)(1-x^8)(1-x^10)).
%Y A029298 Adjacent sequences: A029295 A029296 A029297 this_sequence A029299 A029300 A029301
%Y A029298 Sequence in context: A025853 A025847 A092130 this_sequence A059835 A093998 A029389
%K A029298 nonn
%O A029298 0,7
%A A029298 njas

%I A059835
%S A059835 0,1,2,0,1,2,1,2,0,1,2,2,0,1,2,1,2,0,1,2,0,1,2,1,2,0,1,2,2,0,1,2,1,2,0,
%T A059835 1,2,1,2,0,1,2,2,0,1,2,1,2,0,1,2,0,1,2,1,2,0,1,2,2,0,1,2,1,2,0,1,2,2,0,
%U A059835 1,2,1,2,0,1,2,0,1,2,1,2,0,1,2,2,0,1,2,1,2,0,1,2,1,2,0,1,2,2,0,1,2,1,2
%N A059835 Form triangle as follows: start with three single digits: 0, 1, 2. Each succeeding row is a concatenation of the previous three rows.
%D A059835 C. Pickover, Wonders of Numbers, Oxford University Press, NY, 2001, p. 273.
%H A059835 C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," <a href="http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?first=1&maxdocs=3&type=html&an=0983.00008&format=complete">Zentralblatt review</a>
%e A059835 Triangle begins
%e A059835 0
%e A059835 1
%e A059835 2
%e A059835 0 1 2
%e A059835 1 2 0 1 2
%e A059835 2 0 1 2 1 2 0 1 2
%Y A059835 Cf. A059832.
%Y A059835 Adjacent sequences: A059832 A059833 A059834 this_sequence A059836 A059837 A059838
%Y A059835 Sequence in context: A025847 A092130 A029298 this_sequence A093998 A029389 A025835
%K A059835 easy,nonn,tabf
%O A059835 0,3
%A A059835 Jason Earls (jcearls(AT)cableone.net), Feb 25 2001
%E A059835 More terms from Larry Reeves (larryr(AT)acm.org), Feb 26 2001

%I A093998
%S A093998 1,0,0,1,1,1,1,1,1,1,1,2,0,1,2,1,2,1,2,1,0,2,2,2,2,1,2,2,1,3,1,1,2,1,2,
%T A093998 2,2,3,1,2,3,1,3,2,2,3,1,3,2,1,3,2,2,2,1,2,2,2,4,1,3,3,1,4,3,3,3,1,3,3,
%U A093998 2,4,2,3,3,1,4,2,2,4,2,3,3,2,3,2,2,3,0,2,3,2,4,2,4,3,1,5,3,3,4,2,4,4,3
%N A093998 Number of partitions of n with an even number of distinct Fibonacci parts.
%F A093998 G.f.: (prod_{k, 2..infty} (1 + x^{F_k}) + prod_{k=2..infty} (1 - x^{F_k}))/2
%t A093998 Take[ CoefficientList[ Expand[ Product[1 + x^Fibonacci[k], {k, 2, 13}]/2 + Product[1 - x^Fibonacci[k], {k, 2, 13}]/2], x], 105] (from Robert G. Wilson v May 29 2004)
%Y A093998 Cf. A000119.
%Y A093998 Adjacent sequences: A093995 A093996 A093997 this_sequence A093999 A094000 A094001
%Y A093998 Sequence in context: A092130 A029298 A059835 this_sequence A029389 A025835 A029277
%K A093998 easy,nonn
%O A093998 0,12
%A A093998 Naoki Sato (nsato7(AT)yahoo.ca), May 24 2004
%E A093998 Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), May 29 2004

%I A029389
%S A029389 1,0,0,0,0,1,1,0,0,1,1,1,2,0,1,2,1,2,3,1,2,3,2,3,5,2,3,
%T A029389 5,3,5,7,3,5,7,5,7,10,5,7,10,7,10,13,7,10,14,10,13,17,10,
%U A029389 14,18,13,17,22,14,18,23,17,22,28,18,23,29,22,28,34,23
%N A029389 Expansion of 1/((1-x^5)(1-x^6)(1-x^9)(1-x^12)).
%Y A029389 Adjacent sequences: A029386 A029387 A029388 this_sequence A029390 A029391 A029392
%Y A029389 Sequence in context: A029298 A059835 A093998 this_sequence A025835 A029277 A077905
%K A029389 nonn
%O A029389 0,13
%A A029389 njas

%I A025835
%S A025835 1,0,0,1,0,1,2,0,1,2,1,2,3,1,2,4,2,3,5,2,4,6,3,5,7,4,6,
%T A025835 8,5,7,10,6,8,11,7,10,13,8,11,14,10,13,16,11,14,18,13,16,
%U A025835 20,14,18,22,16,20,24,18,22,26,20,24,29,22,26,31,24,29
%N A025835 Expansion of 1/((1-x^3)(1-x^5)(1-x^6)).
%Y A025835 Adjacent sequences: A025832 A025833 A025834 this_sequence A025836 A025837 A025838
%Y A025835 Sequence in context: A059835 A093998 A029389 this_sequence A029277 A077905 A131331
%K A025835 nonn
%O A025835 0,7
%A A025835 njas

%I A029277
%S A029277 1,0,0,1,0,1,2,0,1,2,1,2,4,1,2,5,2,4,7,2,5,8,4,7,11,5,8,
%T A029277 13,7,11,17,8,13,19,11,17,24,13,19,27,17,24,33,19,27,37,
%U A029277 24,33,44,27,37,49,33,44,57,37,49,63,44,57,73,49,63,80
%N A029277 Expansion of 1/((1-x^3)(1-x^5)(1-x^6)(1-x^12)).
%Y A029277 Adjacent sequences: A029274 A029275 A029276 this_sequence A029278 A029279 A029280
%Y A029277 Sequence in context: A093998 A029389 A025835 this_sequence A077905 A131331 A020513
%K A029277 nonn
%O A029277 0,7
%A A029277 njas

%I A077905
%S A077905 1,0,1,1,0,2,0,1,2,1,3,0,0,4,3,4,1,3,8,6,4,5,10,15,9,0,16,24,25,8,15,41,
%T A077905 48,34,8,55,90,81,27,64,144,172,107,36,209,315,280,70,244,525,594,351,175,
%U A077905 768,1120,944,177,944,1887,2065,1120,766,2832,3951,3186,353,3597,6784,7136
%V A077905 1,0,1,1,0,2,0,1,2,-1,3,0,0,4,-3,4,1,-3,8,-6,4,5,-10,15,-9,0,16,-24,25,-8,-15,41,
%W A077905 -48,34,8,-55,90,-81,27,64,-144,172,-107,-36,209,-315,280,-70,-244,525,-594,351,175,
%X A077905 -768,1120,-944,177,944,-1887,2065,-1120,-766,2832,-3951,3186,-353,-3597,6784,-7136
%N A077905 Expansion of (1-x)^(-1)/(1+x-x^3).
%Y A077905 Adjacent sequences: A077902 A077903 A077904 this_sequence A077906 A077907 A077908
%Y A077905 Sequence in context: A029389 A025835 A029277 this_sequence A131331 A020513 A029276
%K A077905 sign
%O A077905 0,6
%A A077905 njas, Nov 17 2002

%I A131331
%S A131331 1,0,1,1,0,1,1,2,0,1,2,1,3,0,1,3,4,1,4,0,1,5,4,7,1,5,0,1,8,9,5,11,1,6,0,
%T A131331 1,13,12,16,6,16,1,7,0,1,21,22,17,27,7,22,1,8,0,1
%V A131331 1,0,1,1,0,1,-1,2,0,1,2,-1,3,0,1,-3,4,-1,4,0,1,5,-4,7,-1,5,0,1,-8,9,-5,11,-1,6,0,1,13,
%W A131331 -12,16,-6,16,-1,7,0,1,-21,22,-17,27,-7,22,-1,8,0,1
%N A131331 A046854 * A000012(signed).
%C A131331 Row sums = A094967: (1, 1, 2, 2, 5, 5, 13, 13, 34, 34...).
%F A131331 A046854 * A000012(signed by columns, + - + -...).
%e A131331 First few rows of the triangle are:
%e A131331 1;
%e A131331 0, 1;
%e A131331 1, 0, 1;
%e A131331 -1, 2, 0, 1;
%e A131331 2, -1, 3, 0, 1;
%e A131331 -3, 4, -1, 4, 0, 1;
%e A131331 5, -4, 7, -1, 5, 0, 1;
%e A131331 -8, 9, -5, 11, -1, 6, 0, 1;
%e A131331 ...
%Y A131331 Cf. A046854, A000012, A094967.
%Y A131331 Adjacent sequences: A131328 A131329 A131330 this_sequence A131332 A131333 A131334
%Y A131331 Sequence in context: A025835 A029277 A077905 this_sequence A020513 A029276 A109248
%K A131331 nonn,tabl
%O A131331 0,8
%A A131331 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 29 2007

%I A020513
%S A020513 1,2,0,1,2,1,3,1,2,1,5,1,1,1,7,1,2,1,3,1,1,1,11,1,1,1,13,
%T A020513 1,1,1,1,1,2,1,17,1,1,1,19,1,1,1,1,1,1,1,23,1,1,1,5,1,1,
%U A020513 1,3,1,1,1,29,1,1,1,31,1,2
%V A020513 -1,-2,0,1,2,1,3,1,2,1,5,1,1,1,7,1,2,1,3,1,1,1,11,1,1,1,13,
%W A020513 1,1,1,1,1,2,1,17,1,1,1,19,1,1,1,1,1,1,1,23,1,1,1,5,1,1,
%X A020513 1,3,1,1,1,29,1,1,1,31,1,2
%N A020513 Cyclotomic polynomials at x=-1.
%F A020513 For n >= 3: if n = 2*p^m with a prime p then a(n) = p otherwise a(n) = 1. - Ola Veshta (olaveshta(AT)my-deja.com), Jun 01 2001
%p A020513 with(numtheory,cyclotomic); f := n->subs(x=-1,cyclotomic(n,x)); seq(f(i),i=0..64);
%Y A020513 Adjacent sequences: A020510 A020511 A020512 this_sequence A020514 A020515 A020516
%Y A020513 Sequence in context: A029277 A077905 A131331 this_sequence A029276 A109248 A131866
%K A020513 sign
%O A020513 0,2
%A A020513 Simon Plouffe (plouffe(AT)math.uqam.ca)

%I A029276
%S A029276 1,0,0,1,0,1,2,0,1,2,1,3,3,1,3,4,3,5,5,3,6,7,6,8,8,7,10,
%T A029276 11,10,12,13,12,15,17,15,18,20,18,22,24,22,26,28,26,31,
%U A029276 33,31,36,38,36,42,44,42,48,50,49,55,57,56,62,65,64,70
%N A029276 Expansion of 1/((1-x^3)(1-x^5)(1-x^6)(1-x^11)).
%Y A029276 Adjacent sequences: A029273 A029274 A029275 this_sequence A029277 A029278 A029279
%Y A029276 Sequence in context: A077905 A131331 A020513 this_sequence A109248 A131866 A036862
%K A029276 nonn
%O A029276 0,7
%A A029276 njas

%I A109248
%S A109248 1,1,1,2,0,1,2,1,3,3,4,6,7,10,13,17,23,30,40,53,70,93,123,163,216,286,
%T A109248 379,502,665,881,1167,1546,2048,2713,3594,4761,6307,8355,11068,14662,19423,
%U A109248 25730,34085,45153,59815,79238,104968,139053,184206,244021,323259,428227,567280
%V A109248 1,-1,-1,-2,0,-1,2,-1,3,-3,4,-6,7,-10,13,-17,23,-30,40,-53,70,-93,123,-163,216,-286,
%W A109248 379,-502,665,-881,1167,-1546,2048,-2713,3594,-4761,6307,-8355,11068,-14662,19423,
%X A109248 -25730,34085,-45153,59815,-79238,104968,-139053,184206,-244021,323259,-428227,567280
%N A109248 Expansion of (1-x-2x^2)/(1-x^2+x^3).
%C A109248 Diagonal sums of Riordan array (1-x-2x^2,x(1-x)), A109246.
%F A109248 a(n)=a(n-2)-a(n-3)
%Y A109248 Adjacent sequences: A109245 A109246 A109247 this_sequence A109249 A109250 A109251
%Y A109248 Sequence in context: A131331 A020513 A029276 this_sequence A131866 A036862 A094238
%K A109248 easy,sign
%O A109248 0,4
%A A109248 Paul Barry (pbarry(AT)wit.ie), Jun 23 2005

%I A131866
%S A131866 0,2,0,1,2,1,4,3,0,1,3,2,1,2,3,3,0,2,6,7,6,2,1,5,7,4,1,4,5,6,9,7,6,5,6,
%T A131866 10,6,3,2,0,1,2,8,11,10,3,2,1,1,2,11,11,10,8,3,0,8,9,13,11,9,2,5,6,7,9,
%U A131866 10,13,12,11,10,8,7,6,4,1,10,12,9,7,3,2,3,6,9,11,15,11,2,0,2,6,9,10,12
%N A131866 Distance of n-th semiprime to nearest square.
%C A131866 This to semiprimes A001358 as A047972 is to primes A000040.
%C A131866 For each semiprime, find the closest square (preceding or succeeding); subtract, take absolute value.
%F A131866 a(n)=A053188(A001358(n)) (corrected by R. J. Mathar, Nov 19 2007).
%e A131866 a(1) = 0 because the 1st semiprime is 4, which is a square.
%e A131866 a(2) = 2 because the 2nd semiprime is 6, and |6-4| = 2 where 4 is the nearest square to 6.
%e A131866 a(3) = 0 because the 3rd semiprime is 9, which is a square.
%e A131866 a(4) = 1 because the 4th semiprime is 10, and |10-9| = 1 where 9 is the nearest square to 10.
%Y A131866 Cf. A001358, A047972, A053188.
%Y A131866 Adjacent sequences: A131863 A131864 A131865 this_sequence A131867 A131868 A131869
%Y A131866 Sequence in context: A020513 A029276 A109248 this_sequence A036862 A094238 A127793
%K A131866 easy,nonn
%O A131866 1,2
%A A131866 Jonathan Vos Post (jvospost2(AT)yahoo.com), Oct 04 2007
%E A131866 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 24 2007

%I A036862
%S A036862 0,0,1,0,0,1,0,1,2,0,1,2,1,4,3,2,4,5,8,9,6,9,13,18,24,15,22,35,36,59,
%T A036862 37,47,85,78,125,91,101,187,173,255,201,221,386,369,508
%N A036862 Number of partitions satisfying either one of the two conditions cn(0,5) = cn(1,5) <= cn(2,5) = cn(4,5) <= cn(3,5) or cn(2,5) = cn(4,5) <= cn(0,5) = cn(1,5) <= cn(3,5).
%C A036862 For a given partition cn(i,n) means the number of its parts equal to i modulo n.
%C A036862 Short: (0 = 1 <= 2 = 4 <= 3) or (2 = 4 <= 0 = 1 <= 3).
%Y A036862 Adjacent sequences: A036859 A036860 A036861 this_sequence A036863 A036864 A036865
%Y A036862 Sequence in context: A029276 A109248 A131866 this_sequence A094238 A127793 A127771
%K A036862 nonn,more
%O A036862 1,9
%A A036862 Olivier Gerard (ogerard(AT)ext.jussieu.fr)

%I A094238
%S A094238 2,0,1,2,1,7,1,2,5,5,6,8,2,5,9,5,1,5,0,6,3,7,4,7,7,8,9,7,7,1,7,8,7,6,8,
%T A094238 1,2,3,1,2,1,5,0,5,8,3,2,0,2,6,7,3,7,7,1,2,1,0,5,4,9,7,9,1,1,5,9,3,2,5,
%U A094238 2,9,7,5,3,8,1,1,0,8,7,3,6,2,4,6,7,3,8,8,0,2,8,5,1,2,5,3,7,4,0,1,8,4,2
%N A094238 Decimal expansion of (pi+e)^3.
%e A094238 201.21712556825951506374778977
%Y A094238 Cf. A001113, A000796, A059742.
%Y A094238 Adjacent sequences: A094235 A094236 A094237 this_sequence A094239 A094240 A094241
%Y A094238 Sequence in context: A109248 A131866 A036862 this_sequence A127793 A127771 A118407
%K A094238 cons,nonn
%O A094238 3,1
%A A094238 Mohammad K. Azarian (azarian(AT)evansville.edu), May 30 2004

%I A127793
%S A127793 1,0,1,0,1,2,0,1,2,2,0,0,0,2,3,0,1,2,0,3,3,0,0,0,0,0,3,4,0,1,2,2,0,0,4,
%T A127793 4,0,0,0,0,0,0,0,4,5,0,0,0,2,3,0,0,0,5,5,0,0,0,0,0,0,0,0,0,5,6
%V A127793 1,0,1,0,-1,2,0,1,-2,2,0,0,0,-2,3,0,-1,2,0,-3,3,0,0,0,0,0,-3,4,0,1,-2,2,0,0,-4,4,0,0,0,
%W A127793 0,0,0,0,-4,5,0,0,0,-2,3,0,0,0,-5,5,0,0,0,0,0,0,0,0,0,-5,6
%N A127793 Inverse of number triangle A(n,k)=if(k<=n,if(n<=2k,1/floor((n+2)/2)),0),0).
%C A127793 It is conjectured that the triangle is an integer triangle. The triangle and its inverse both appear to have row sums equal to the all 1's sequence.
%e A127793 Triangle begins
%e A127793 1,
%e A127793 0, 1,
%e A127793 0, -1, 2,
%e A127793 0, 1, -2, 2,
%e A127793 0, 0, 0, -2, 3,
%e A127793 0, -1, 2, 0, -3, 3,
%e A127793 0, 0, 0, 0, 0, -3, 4,
%e A127793 0, 1, -2, 2, 0, 0, -4, 4,
%e A127793 0, 0, 0, 0, 0, 0, 0, -4, 5,
%e A127793 0, 0, 0, -2, 3, 0, 0, 0, -5, 5,
%e A127793 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 6,
%e A127793 0, -1, 2, 0, -3, 3, 0, 0, 0, 0, -6, 6,
%e A127793 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 7
%e A127793 Inverse of the triangle
%e A127793 1,
%e A127793 0, 1,
%e A127793 0, 1/2, 1/2,
%e A127793 0, 0, 1/2, 1/2,
%e A127793 0, 0, 1/3, 1/3, 1/3,
%e A127793 0, 0, 0, 1/3, 1/3, 1/3,
%e A127793 0, 0, 0, 1/4, 1/4, 1/4, 1/4,
%e A127793 0, 0, 0, 0, 1/4, 1/4, 1/4, 1/4,
%e A127793 0, 0, 0, 0, 1/5, 1/5, 1/5, 1/5, 1/5,
%e A127793 0, 0, 0, 0, 0, 1/5, 1/5, 1/5, 1/5, 1/5, 0;
%e A127793 0, 0, 0, 0, 0, 1/6, 1/6, 1/6, 1/6, 1/6, 1/6
%Y A127793 Adjacent sequences: A127790 A127791 A127792 this_sequence A127794 A127795 A127796
%Y A127793 Sequence in context: A131866 A036862 A094238 this_sequence A127771 A118407 A101663
%K A127793 sign,tabl
%O A127793 0,6
%A A127793 Paul Barry (pbarry(AT)wit.ie), Jan 29 2007

%I A127771
%S A127771 1,0,1,0,1,2,0,1,2,2,0,0,0,2,4,0,1,2,0,4,2,0,0,0,0,0,2,6,0,1,2,2,0,0,6,
%T A127771 4,0,0,0,0,0,0,0,4,6,0,0,0,2,4,0,0,0,6,4,0,0,0,0,0,0,0,0,0,4,10,0,1,2,0,
%U A127771 4,2,0,0,0,0,10
%V A127771 1,0,1,0,-1,2,0,1,-2,2,0,0,0,-2,4,0,-1,2,0,-4,2,0,0,0,0,0,-2,6,0,1,-2,2,0,0,-6,4,0,0,0,
%W A127771 0,0,0,0,-4,6,0,0,0,-2,4,0,0,0,-6,4,0,0,0,0,0,0,0,0,0,-4,10,0,-1,2,0,-4,2,0,0,0,0,-10
%N A127771 Inverse of number triangle A(n,k)=if(k<=n,if(n<=2k,1/Euler_phi(n+1),0),0).
%C A127771 It is conjectured that all elements of the triangle are integers. Row sums are A127772.
%e A127771 Triangle begins
%e A127771 1,
%e A127771 0, 1,
%e A127771 0, -1, 2,
%e A127771 0, 1, -2, 2,
%e A127771 0, 0, 0, -2, 4,
%e A127771 0, -1, 2, 0, -4, 2,
%e A127771 0, 0, 0, 0, 0, -2, 6,
%e A127771 0, 1, -2, 2, 0, 0, -6, 4,
%e A127771 0, 0, 0, 0, 0, 0, 0, -4, 6,
%e A127771 0, 0, 0, -2, 4, 0, 0, 0, -6, 4,
%e A127771 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 10,
%e A127771 0, -1, 2, 0, -4, 2, 0, 0, 0, 0, -10, 4,
%e A127771 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 12
%e A127771 Inverse of the triangle
%e A127771 1,
%e A127771 0, 1,
%e A127771 0, 1/2, 1/2,
%e A127771 0, 0, 1/2, 1/2,
%e A127771 0, 0, 1/4, 1/4, 1/4,
%e A127771 0, 0, 0, 1/2, 1/2, 1/2,
%e A127771 0, 0, 0, 1/6, 1/6, 1/6, 1/6,
%e A127771 0, 0, 0, 0, 1/4, 1/4, 1/4, 1/4,
%e A127771 0, 0, 0, 0, 1/6, 1/6, 1/6, 1/6, 1/6,
%e A127771 0, 0, 0, 0, 0, 1/4, 1/4, 1/4, 1/4, 1/4,
%e A127771 0, 0, 0, 0, 0, 1/10, 1/10, 1/10, 1/10, 1/10, 1/10
%Y A127771 Adjacent sequences: A127768 A127769 A127770 this_sequence A127772 A127773 A127774
%Y A127771 Sequence in context: A036862 A094238 A127793 this_sequence A118407 A101663 A062169
%K A127771 sign,tabl
%O A127771 0,6
%A A127771 Paul Barry (pbarry(AT)wit.ie), Jan 28 2007

%I A118407
%S A118407 1,0,1,2,0,1,2,2,0,1,0,2,2,0,1,2,0,2,2,0,1,4,2,0,2,2,0,1,6,4,2,0,2,2,0,
%T A118407 1,4,6,4,2,0,2,2,0,1,6,4,6,4,2,0,2,2,0,1,20,6,4,6,4,2,0,2,2,0,1,26,20,6,
%U A118407 4,6,4,2,0,2,2,0,1,12,26,20,6,4,6,4,2,0,2,2,0,1
%V A118407 1,0,1,-2,0,1,2,-2,0,1,0,2,-2,0,1,-2,0,2,-2,0,1,4,-2,0,2,-2,0,1,-6,4,-2,0,2,-2,0,1,4,
%W A118407 -6,4,-2,0,2,-2,0,1,6,4,-6,4,-2,0,2,-2,0,1,-20,6,4,-6,4,-2,0,2,-2,0,1,26,-20,6,4,-6,4,
%X A118407 -2,0,2,-2,0,1,-12,26,-20,6,4,-6,4,-2,0,2,-2,0,1
%N A118407 Triangle, read by rows, equal to the matrix square of triangle A118404; also equals the matrix inverse of triangle A118401.
%C A118407 This triangle has an integer matrix square-root (A118404) if the main diagonal of the square-root is allowed to be signed. Even though the columns of this triangle are all the same, the columns of the matrix square-root A118404 are all different.
%F A118407 G.f.: A(x,y) = (1+x)^2/(1+x^2)/(1+2*x+2*x^2)/(1-x*y). Column g.f.: (1+x)^2/(1+x^2)/(1+2*x+2*x^2).
%e A118407 Triangle begins:
%e A118407 1;
%e A118407 0, 1;
%e A118407 -2, 0, 1;
%e A118407 2,-2, 0, 1;
%e A118407 0, 2,-2, 0, 1;
%e A118407 -2, 0, 2,-2, 0, 1;
%e A118407 4,-2, 0, 2,-2, 0, 1;
%e A118407 -6, 4,-2, 0, 2,-2, 0, 1;
%e A118407 4,-6, 4,-2, 0, 2,-2, 0, 1;
%e A118407 6, 4,-6, 4,-2, 0, 2,-2, 0, 1;
%e A118407 -20, 6, 4,-6, 4,-2, 0, 2,-2, 0, 1;
%e A118407 26,-20, 6, 4,-6, 4,-2, 0, 2,-2, 0, 1; ...
%o A118407 (PARI) {T(n,k)=polcoeff(polcoeff((1+x)^2/(1+x^2)/(1+2*x+2*x^2)/(1-x*y+x*O(x^n)),n,x)+y*O(y^k),k,y)}
%Y A118407 Cf. A118404 (matrix square-root), A118401 (matrix inverse), A118408 (row sums), A118409 (unsigned row sums).
%Y A118407 Adjacent sequences: A118404 A118405 A118406 this_sequence A118408 A118409 A118410
%Y A118407 Sequence in context: A094238 A127793 A127771 this_sequence A101663 A062169 A113680
%K A118407 sign,tabl
%O A118407 0,4
%A A118407 Paul D. Hanna (pauldhanna(AT)juno.com), Apr 27 2006

%I A101663
%S A101663 0,1,0,2,0,1,2,2,0,1,0,2,2,2,2,2,0,1,0,2,0,1,2,2,2,2,2,2,2,2,2,2,0,1,0,
%T A101663 2,0,1,2,2,0,1,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,0,1,0,2,0,1,
%U A101663 2,2,0,1,0,2,2,2,2,2,0,1,0,2,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2
%N A101663 Fixed point of morphism 0 -> 01, 1 -> 02, 2 -> 22.
%F A101663 a(2n) = 2[a(n)==2], a(2n+1) = 1 + [a(n)>0].
%t A101663 Nest[ Function[ l, {Flatten[(l /. {0 -> {0, 1}, 1 -> {0, 2}, 2 -> {2, 2}})]}], {0}, 7] (from Robert G. Wilson v Feb 28 2005)
%Y A101663 Adjacent sequences: A101660 A101661 A101662 this_sequence A101664 A101665 A101666
%Y A101663 Sequence in context: A127793 A127771 A118407 this_sequence A062169 A113680 A128187
%K A101663 nonn,easy
%O A101663 0,4
%A A101663 Ralf Stephan, Dec 11 2004

%I A062169
%S A062169 0,1,0,1,2,0,1,2,2,0,1,2,1,4,0,1,2,0,0,0,0,1,2,6,3,1,6,0,1,2,6,0,0,0,0,
%T A062169 0,1,2,6,6,3,0,0,0,0,1,2,6,4,0,0,0,0,0,0,1,2,6,2,10,5,2,5,1,10,0,1,2,6,
%U A062169 0,0,0,0,0,0,0,0,0,1,2,6,11,3,5,9,7,11,6,1,12,0,1,2,6,10,8,6,0,0,0,0,0
%N A062169 Triangle of n! mod k.
%e A062169 a(7,4) = 4! mod 7 = 24 mod 7 = 3. Rows are: (0), (1,0), (1,2,0), (1,2,2,0), (1,2,1,4,0), (1,2,0,0,0,0), (1,2,6,3,1,6,0) etc.
%Y A062169 First zero in each row is when k=A002034. Maximum value in each row is A062170. Number of distinct values in each row is A038203. Cf. A000142, A048158, A051127.
%Y A062169 Adjacent sequences: A062166 A062167 A062168 this_sequence A062170 A062171 A062172
%Y A062169 Sequence in context: A127771 A118407 A101663 this_sequence A113680 A128187 A133121
%K A062169 nonn,tabl
%O A062169 1,5
%A A062169 Henry Bottomley (se16(AT)btinternet.com), Jun 11 2001

%I A113680
%S A113680 1,0,1,2,0,1,2,2,0,1,2,2,2,0,1,2,2,2,2,0,1,2,2,2,2,2,0,1,2,2,2,2,2,2,0,
%T A113680 1,2,2,2,2,2,2,2,0,1,2,2,2,2,2,2,2,2,0,1,2,2,2,2,2,2,2,2,2,0,1,2,2,2,2,
%U A113680 2,2,2,2,2,2,0,1,2,2,2,2,2,2,2,2
%V A113680 1,0,1,-2,0,1,-2,-2,0,1,-2,-2,-2,0,1,-2,-2,-2,-2,0,1,-2,-2,-2,-2,-2,0,1,-2,-2,-2,-2,-2,
%W A113680 -2,0,1,-2,-2,-2,-2,-2,-2,-2,0,1,-2,-2,-2,-2,-2,-2,-2,-2,0,1,-2,-2,-2,-2,-2,-2,-2,-2,
%X A113680 -2,0,1,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,1,-2,-2,-2,-2,-2,-2,-2,-2
%N A113680 Riordan array ((1-x-2x^2)/(1-x),x).
%C A113680 Inverse of A113678. Sequence array of A113679. Row sums are 3-2n-2*0^n. Diagonal sums are 1-n.
%F A113680 Number triangle T(n, k)=if(k<=n, C(0, n-k)+2*C(1, n-k)-2, 0).
%e A113680 Triangle begins
%e A113680 1;
%e A113680 0, 1;
%e A113680 -2, 0, 1;
%e A113680 -2, -2, 0, 1;
%e A113680 -2, -2, -2, 0, 1;
%e A113680 -2, -2, -2, -2, 0, 1;
%Y A113680 Adjacent sequences: A113677 A113678 A113679 this_sequence A113681 A113682 A113683
%Y A113680 Sequence in context: A118407 A101663 A062169 this_sequence A128187 A133121 A091602
%K A113680 easy,sign,tabl
%O A113680 0,4
%A A113680 Paul Barry (pbarry(AT)wit.ie), Nov 04 2005

%I A128187
%S A128187 1,1,1,2,0,1,2,2,0,1,3,0,1,0,1,3,3,1,1,0,1,4,0,1,0,1,0,1,4,4,1,2,0,1,0,
%T A128187 1,5,0,2,0,1,0,1,0,1,5,5,1,2,1,1,0,1,0,1
%N A128187 A128174 * A051731.
%C A128187 Row sums = A128188: (1, 2, 3, 5, 5, 9, 7, 13, 10, 17,...)
%F A128187 A128174 * A051731 as infinite lower triangular matrices.
%e A128187 First few rows of the triangle are:
%e A128187 1;
%e A128187 1, 1;
%e A128187 2, 0, 1;
%e A128187 2, 2, 0, 1;
%e A128187 3, 0, 1, 0, 1;
%e A128187 3, 3, 1, 1, 0, 1;
%e A128187 4, 0, 1, 0, 1, 0, 1;
%e A128187 4, 4, 1, 2, 0, 1, 0, 1;
%e A128187 ...
%Y A128187 Cf. A128188, A128174, A051731.
%Y A128187 Adjacent sequences: A128184 A128185 A128186 this_sequence A128188 A128189 A128190
%Y A128187 Sequence in context: A101663 A062169 A113680 this_sequence A133121 A091602 A035465
%K A128187 nonn,tabl
%O A128187 1,4
%A A128187 Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 07 2007

%I A133121
%S A133121 1,1,1,2,0,1,2,2,0,1,3,2,1,0,1,4,2,3,1,0,1,5,4,2,2,1,0,1,6,6,3,3,2,1,0,
%T A133121 1,8,7,5,4,2,2,1,0,1,10,8,10,3,5,2,2,1,0,1,12,13,8,9,4,4,2,2,1,0,1,15,
%U A133121 15,14,10,8,5,4,2,2,1,0,1,18,21,15,16,8,9,4,4,2,2,1,0,1,22,25,23,17,17
%N A133121 Triangle T(n,k) read by rows = number of partitions of n such that number of parts minus number of distinct parts is equal to k, k = 0..n-1.
%F A133121 G.f.: Product(1+x^n/(1-y*x^n),n=1..inf).
%e A133121 1
%e A133121 1,1
%e A133121 2,0,1
%e A133121 2,2,0,1
%e A133121 3,2,1,0,1
%e A133121 4,2,3,1,0,1
%e A133121 5,4,2,2,1,0,1
%e A133121 6,6,3,3,2,1,0,1
%e A133121 8,7,5,4,2,2,1,0,1
%e A133121 10,8,10,3,5,2,2,1,0,1
%e A133121 12,13,8,9,4,4,2,2,1,0,1
%e A133121 15,15,14,10,8,5,4,2,2,1,0,1
%e A133121 18,21,15,16,8,9,4,4,2,2,1,0,1
%o A133121 (PARI) partitm(n,m,nmin)={ local(resul,partj) ; if( n < 0 || m <0, return([;]) ; ) ; resul=matrix(0,m); if(m==0, return(resul); ) ; for(j=max(1,nmin),n\m, partj=partitm(n-j,m-1,j) ; for(r1=1,matsize(partj)[1], resul=concat(resul,concat([j],partj[r1,])) ; ) ; ) ; if(m==1 && n >= nmin, resul=concat(resul,[[n]]) ; ) ; return(resul) ; } partit(n)={ local(resul,partm,filr) ; if( n < 0, return([;]) ; ) ; resul=matrix(0,n) ; for(m=1,n, partm=partitm(n,m,1) ; filr=vector(n-m) ; for(r1=1,matsize(partm)[1], resul=concat( resul,concat(partm[r1,],filr) ) ; ) ; ) ; return(resul) ; } A133121row(n)={ local(p=partit(n),resul=vector(n),nprts,ndprts) ; for(r=1,matsize(p)[1], nprts=0 ; ndprts=0 ; for(c=1,n, if( p[r,c]==0, break, nprts++ ; if(c==1, ndprts++, if(p[r,c]!=p[r,c-1], ndprts++ ) ; ) ; ) ; ) ; k=nprts-ndprts; resul[k+1]++ ; ) ; return(resul) ; } A133121()={ for(n=1,20, arow=A133121row(n) ; for(k=1,n, print1(arow[k],",") ; ) ; ) ; } A133121() ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 28 2007
%Y A133121 Cf. A000009, A090858.
%Y A133121 Adjacent sequences: A133118 A133119 A133120 this_sequence A133122 A133123 A133124
%Y A133121 Sequence in context: A062169 A113680 A128187 this_sequence A091602 A035465 A096144
%K A133121 easy,nonn,tabl
%O A133121 1,4
%A A133121 Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 18 2007
%E A133121 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 28 2007

%I A091602
%S A091602 1,1,1,2,0,1,2,2,0,1,3,2,1,0,1,4,3,2,1,0,1,5,4,3,1,1,0,1,6,7,3,3,1,1,0,
%T A091602 1,8,8,6,3,2,1,1,0,1,10,12,7,5,3,2,1,1,0,1,12,15,11,6,5,2,2,1,1,0,1,15,
%U A091602 21,14,10,5,5,2,2,1,1,0,1,18,26,20,12,9,5,4,2,2,1,1,0,1,22,35,25,18,11
%N A091602 Triangle: T(n,k) = number of partitions of n such that some part is repeated k times and no part is repeated more than k times.
%F A091602 G.f.=G=G(t,x)=sum(t^k*(product((1-x^((k+1)j))/(1-x^j), j=1..infinity)-product((1-x^(kj))/(1-x^j), j=1..infinity)), k=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006
%e A091602 1; 1,1; 2,0,1; 2,2,0,1; 3,2,1,0,1; ...
%e A091602 In the partition 5+2+2+2+1+1, 2 is repeated 3 times, no part is repeated more than 3 times.
%p A091602 g:=sum(t^k*(product((1-x^((k+1)*j))/(1-x^j),j=1..50)-product((1-x^(k*j))/(1-x^j),j=1..50)),k=1..50): gser:=simplify(series(g,x=0,20)): for n from 1 to 13 do P[n]:=coeff(gser,x^n) od: for n from 1 to 13 do seq(coeff(P[n],t^j),j=1..n) od; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006
%Y A091602 Row sums: A000041. Inverse: A091603. Square: A091604.
%Y A091602 Columns 1-6: A000009, A091605-A091609. Convergent of columns: A002865.
%Y A091602 Adjacent sequences: A091599 A091600 A091601 this_sequence A091603 A091604 A091605
%Y A091602 Sequence in context: A113680 A128187 A133121 this_sequence A035465 A096144 A118401
%K A091602 nonn,tabl
%O A091602 1,4
%A A091602 Christian G. Bower (bowerc(AT)usa.net), Jan 23 2004

%I A035465
%S A035465 0,0,0,1,0,0,1,1,0,0,1,2,0,1,2,2,0,1,3,3,1,2,4,4,1,3,6,6,2,5,8,7,3,7,
%T A035465 12,10,5,10,15,13,7,15,21,17,11,20,27,22,16,28,36,29,22,37,46,38,31,50,
%U A035465 60,50,42,65,77,64,57,86,99,82,76,111,125,106,101,144,159,135,132
%N A035465 Number of partitions of n into parts 8k+4 or 8k+7.
%Y A035465 Adjacent sequences: A035462 A035463 A035464 this_sequence A035466 A035467 A035468
%Y A035465 Sequence in context: A128187 A133121 A091602 this_sequence A096144 A118401 A113678
%K A035465 nonn
%O A035465 1,12
%A A035465 Olivier Gerard (ogerard(AT)ext.jussieu.fr)

%I A096144
%S A096144 1,1,1,2,0,1,2,2,0,1,4,1,1,0,1,4,3,2,1,0,1,7,3,2,1,1,0,1,8,6,2,3,1,1,0,
%T A096144 1,12,5,6,2,2,1,1,0,1,14,11,5,4,3,2,1,1,0,1,21,11,8,5,4,2,2,1,1,0,1,24,
%U A096144 17,11,9,4,5,2,2,1,1,0,1,34,20,15,9,8,4,4,2,2,1,1,0,1,41,30,18,14,9,7,5
%N A096144 Triangle T(n,k) = number of partitions of n in which the least part occurs exactly k times, k=1..n.
%F A096144 G.f. for k-th column: Sum(x^(k*m)/Product(1-x^i, i=m+1..infinity), m=1..infinity).
%e A096144 1; 1,1; 2,0,1; 2,2,0,1; 4,1,1,0,1; 4,3,2,1,0,1; 7,3,2,1,1,0,1; ....
%Y A096144 Cf. A002865(first column), A096373(second column), A000041(row sums).
%Y A096144 Adjacent sequences: A096141 A096142 A096143 this_sequence A096145 A096146 A096147
%Y A096144 Sequence in context: A133121 A091602 A035465 this_sequence A118401 A113678 A110249
%K A096144 easy,nonn,tabl
%O A096144 1,4
%A A096144 Vladeta Jovovic (vladeta(AT)Eunet.yu), Jul 24 2004

%I A118401
%S A118401 1,0,1,2,0,1,2,2,0,1,4,2,2,0,1,6,4,2,2,0,1,8,6,4,2,2,0,1,10,8,6,4,2,2,0,
%T A118401 1,12,10,8,6,4,2,2,0,1,14,12,10,8,6,4,2,2,0,1,16,14,12,10,8,6,4,2,2,0,1,
%U A118401 18,16,14,12,10,8,6,4,2,2,0,1,20,18,16,14,12,10,8,6,4,2,2,0,1
%V A118401 1,0,1,2,0,1,-2,2,0,1,4,-2,2,0,1,-6,4,-2,2,0,1,8,-6,4,-2,2,0,1,-10,8,-6,4,-2,2,0,1,12,
%W A118401 -10,8,-6,4,-2,2,0,1,-14,12,-10,8,-6,4,-2,2,0,1,16,-14,12,-10,8,-6,4,-2,2,0,1,-18,16,
%X A118401 -14,12,-10,8,-6,4,-2,2,0,1,20,-18,16,-14,12,-10,8,-6,4,-2,2,0,1
%N A118401 Triangle, read by rows, equal to the matrix square of triangle A118400; also equals the matrix inverse of triangle A118407.
%C A118401 This triangle has an integer matrix square-root (A118400) if the main diagonal of the square-root is allowed to be signed. Even though the columns of this triangle are all the same, the columns of the matrix square-root A118400 are all different.
%F A118401 G.f.: A(x,y) = (1 + 2*x + 2*x^2)*(1+x^2)/(1+x)^2/(1-x*y). Column g.f.: (1 + 2*x + 2*x^2)*(1+x^2)/(1+x)^2.
%e A118401 Triangle begins:
%e A118401 1;
%e A118401 0, 1;
%e A118401 2, 0, 1;
%e A118401 -2, 2, 0, 1;
%e A118401 4,-2, 2, 0, 1;
%e A118401 -6, 4,-2, 2, 0, 1;
%e A118401 8,-6, 4,-2, 2, 0, 1;
%e A118401 -10, 8,-6, 4,-2, 2, 0, 1;
%e A118401 12,-10, 8,-6, 4,-2, 2, 0, 1;
%e A118401 -14, 12,-10, 8,-6, 4,-2, 2, 0, 1;
%e A118401 16,-14, 12,-10, 8,-6, 4,-2, 2, 0, 1; ...
%o A118401 (PARI) {T(n,k)=polcoeff(polcoeff((1+2*x+2*x^2)*(1+x^2)/(1+x)^2/(1-x*y+x*O(x^n)),n,x)+y*O(y^k),k,y)}
%Y A118401 Cf. A118400 (matrix square-root), A118402 (row sums), A118403 (unsigned row sums), A118407 (matrix inverse).
%Y A118401 Adjacent sequences: A118398 A118399 A118400 this_sequence A118402 A118403 A118404
%Y A118401 Sequence in context: A091602 A035465 A096144 this_sequence A113678 A110249 A067460
%K A118401 sign,tabl
%O A118401 0,4
%A A118401 Paul D. Hanna (pauldhanna(AT)juno.com), Apr 27 2006

%I A113678
%S A113678 1,0,1,2,0,1,2,2,0,1,6,2,2,0,1,10,6,2,2,0,1,22,10,6,2,2,0,1,42,22,10,6,
%T A113678 2,2,0,1,86,42,22,10,6,2,2,0,1,170,86,42,22,10,6,2,2,0,1,342,170,86,42,
%U A113678 22,10,6,2,2,0,1,682,342,170,86,42,22,10,6,2,2,0,1,1366,682,342,170,86
%N A113678 Sequence array for A078008.
%C A113678 Row sums are A001045(n+1). Diagonal sums are A053088. Inverse is A113680.
%F A113678 Riordan array ((1-x)/(1-x-2x^2), x); Number triangle T(n, k)=if(k<=n, (2^(n-k)+2(-1)^(n-k))/3, 0); T(n, k)=sum{i=0..n, C(n-i, k)C(k, n-i)(2^i+2(-1)^i)/3}.
%e A113678 Triangle begins
%e A113678 1;
%e A113678 0, 1;
%e A113678 2, 0, 1;
%e A113678 2, 2, 0, 1;
%e A113678 6, 2, 2, 0, 1;
%e A113678 10, 6, 2, 2, 0, 1;
%e A113678 22, 10, 6, 2, 2, 0, 1;
%Y A113678 Adjacent sequences: A113675 A113676 A113677 this_sequence A113679 A113680 A113681
%Y A113678 Sequence in context: A035465 A096144 A118401 this_sequence A110249 A067460 A128256
%K A113678 easy,nonn,tabl
%O A113678 0,4
%A A113678 Paul Barry (pbarry(AT)wit.ie), Nov 04 2005

%I A110249
%S A110249 1,0,1,1,2,0,1,2,2,0,2,3,5,0,3,3,2,0,1,4,4,0,0,5,11,0,3,4,2,0,3,2,4,0,4,
%T A110249 7,2,0,4,5,14,0,2,6,6,0,2,8,2,0,7,5,4,0,3,8,4,0,2,10,28,0,1,5,6,0,2,2,4,
%U A110249 0,4,10,4,0,7,7,0,0,3,10,8,0,4,11,39,0,7,2,2,0,5,8,8,0,2,13,2,0,5,9,6
%N A110249 Diagonal sums of a Jacobi number triangle.
%C A110249 Diagonal sums of number triangle A110247.
%F A110249 a(n)=sum{k=0..floor(n/2), Jacobi(k, 2n-4k+1)}
%Y A110249 Adjacent sequences: A110246 A110247 A110248 this_sequence A110250 A110251 A110252
%Y A110249 Sequence in context: A096144 A118401 A113678 this_sequence A067460 A128256 A065051
%K A110249 easy,nonn
%O A110249 0,5
%A A110249 Paul Barry (pbarry(AT)wit.ie), Jul 17 2005

%I A067460
%S A067460 2,0,1,2,2,1,1,1,2,1,0,1,1,0,2,1,2,1,0,0,1,0,2,1,1,1,0,2,1,1,1,0,1,0,1,
%T A067460 1,1,1,2,1,2,1,0,1,1,1,2,0,2,1,1,0,1,1,1,2,1,1,1,1,0,1,1,0,1,1,2,1,2,1,
%U A067460 1,2,0,1,1,2,1,1,1,1,0,1,0,1,0,0,1,1,1,2,2,2,1,1,0,2,1,1,1,1,2,1,2,1,2
%N A067460 mu(prime(n)-1)+1.
%Y A067460 Adjacent sequences: A067457 A067458 A067459 this_sequence A067461 A067462 A067463
%Y A067460 Sequence in context: A118401 A113678 A110249 this_sequence A128256 A065051 A084665
%K A067460 easy,nonn
%O A067460 1,1
%A A067460 Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 23 2002

%I A128256
%S A128256 1,1,1,2,0,1,2,2,1,1,3,0,3,2,1,3,3,3,5,3,1,4,0,6,8,8,4,1,4,4,6,14,16,12,
%T A128256 5,1,5,0,10,20,30,28,17,6,1,5,5,10,30,50,58,45,23,7,1
%V A128256 1,-1,1,2,0,1,-2,2,1,1,3,0,3,2,1,-3,3,3,5,3,1,4,0,6,8,8,4,1,-4,4,6,14,16,12,5,1,5,0,10,
%W A128256 20,30,28,17,6,1,-5,5,10,30,50,58,45,23,7,1
%N A128256 A004736(signed) * A007318.
%C A128256 Row sums = A053088: (1, 0, 3, 2, 9, 12, 31, 54, 117,...)
%F A128256 A004736(with alternate signs: (1; -2,1; 3,-2,1;...) * A007318, Pascal's triangle.
%e A128256 First few rows of the triangle are:
%e A128256 1;
%e A128256 -1, 1;
%e A128256 2, 0, 1;
%e A128256 -2, 2, 1, 1;
%e A128256 3, 0, 3, 2, 1;
%e A128256 -3, 3, 3, 5, 3, 1;
%e A128256 4, 0, 6, 8, 8, 4, 1;
%e A128256 -4, 4, 6, 14, 16, 12, 5, 1;
%e A128256 5, 0, 10, 10, 30, 28, 17, 6, 1;
%e A128256 ...
%Y A128256 Cf. A004736, A007318, A053088.
%Y A128256 Adjacent sequences: A128253 A128254 A128255 this_sequence A128257 A128258 A128259
%Y A128256 Sequence in context: A113678 A110249 A067460 this_sequence A065051 A084665 A035392
%K A128256 tabl,sign
%O A128256 1,4
%A A128256 Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 21 2007

%I A065051
%S A065051 1,1,2,0,1,2,2,1,2,1,2,0,1,0,1,0,1,2,3,2,3,1,0,1,0,1,0,1,0,1,0,1,2,
%T A065051 3,2,3,2,1,2,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,2,
%U A065051 3,2,3,2,3,2,3,2,3,2,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0
%N A065051 Let R(n) = n-th term of Recaman's sequence A005132; write R(n) = q*n + r with 0 <= r < n; sequence gives values of q.
%H A065051 <a href="http://www.research.att.com/~njas/sequences/Sindx_Rea.html#Recaman">Index entries for sequences related to Recaman's sequence</a>
%Y A065051 Cf. A065052, A005132.
%Y A065051 Adjacent sequences: A065048 A065049 A065050 this_sequence A065052 A065053 A065054
%Y A065051 Sequence in context: A110249 A067460 A128256 this_sequence A084665 A035392 A007149
%K A065051 nonn
%O A065051 1,3
%A A065051 Allan Wilks, Nov 06 2001

%I A084665
%S A084665 2,0,1,2,2,1,2,2,6,5,6,1,4,1,2,1,4,1,16,4,4,1,4,2,10,4,4,1,6,3,18,3,2,5,
%T A084665 14,4,8,2,2,2,14,1,6,7,8,3,2,4,14,3,2,1,6,7,2,5,4,2,12,2,16,3,6,12,6,3,
%U A084665 12,4,4,4,4,7,4,7,6,4,6,5,2,15,16,1,4,3,4,13,6,6,12,2,10,7,2,1,4,1,4,3
%N A084665 A074104(n)/n.
%Y A084665 Cf. A074104.
%Y A084665 Adjacent sequences: A084662 A084663 A084664 this_sequence A084666 A084667 A084668
%Y A084665 Sequence in context: A067460 A128256 A065051 this_sequence A035392 A007149 A028832
%K A084665 nonn
%O A084665 1,1
%A A084665 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 03 2003

%I A035392
%S A035392 0,0,1,0,0,1,1,0,1,2,0,1,2,2,1,2,4,1,2,5,4,2,5,8,2,5,10,7,5,11,14,5,11,
%T A035392 19,12,11,21,24,11,22,33,22,22,38,41,22,40,58,37,41,68,67,41,73,95,63,
%U A035392 75,114,108,76,124,155,106,129,188,173,131,208,246,174,218,303,271
%N A035392 Number of partitions of n into parts 7k or 7k+3.
%Y A035392 Adjacent sequences: A035389 A035390 A035391 this_sequence A035393 A035394 A035395
%Y A035392 Sequence in context: A128256 A065051 A084665 this_sequence A007149 A028832 A033773
%K A035392 nonn
%O A035392 1,10
%A A035392 Olivier Gerard (ogerard(AT)ext.jussieu.fr)

%I A007149 M0017
%S A007149 0,0,1,2,0,1,2,2,1,2,4,3,4,4,5,5,4,5,8,6,8,7,8,8,9,9,10,10,15,11,12,12,
%T A007149 11,12,16,13,16,14,15,15,17,16,17,17,19,18,19,19,20,20,21,21,23,22,23,23
%N A007149 2-part of number of graphs on n nodes.
%D A007149 Steven C. Cater and Robert W. Robinson, "Exponents of 2 in the numbers of unlabeled graphs and tournaments," Congressus Numerantium, 82 (1991), pp. 139-155.
%Y A007149 Power of 2 dividing A000088.
%Y A007149 Adjacent sequences: A007146 A007147 A007148 this_sequence A007150 A007151 A007152
%Y A007149 Sequence in context: A065051 A084665 A035392 this_sequence A028832 A033773 A029275
%K A007149 nonn
%O A007149 0,4
%A A007149 njas

%I A028832
%S A028832 0,1,2,0,1,2,2,2,0,1,2,2,2,3,2,0,1,2,4,2,3,4,3,2,0,1,2,3,3,2,4,2,3,3,
%T A028832 2,0,1,2,2,2,2,2,4,3,3,5,3,2,0,1,2,4,3,4,2,2,3,2,4,3,5,3,2,0,1,2,5,2,
%U A028832 4,3,4,2,3,2,2,5,4,3,3,2,0,1,2,2,3,4,2,3,3,2,3,4,4,6,3,3,3,3,2,0,1,2
%N A028832 Number of distinct integers in period of continued fraction for sqrt(n), or 0 if n = square.
%Y A028832 Adjacent sequences: A028829 A028830 A028831 this_sequence A028833 A028834 A028835
%Y A028832 Sequence in context: A084665 A035392 A007149 this_sequence A033773 A029275 A058739
%K A028832 nonn
%O A028832 1,3
%A A028832 Olivier Gerard (ogerard(AT)ext.jussieu.fr)

%I A033773
%S A033773 1,1,1,2,0,1,2,2,2,2,3,0,4,3,2,4,3,4,1,4,4,6,6,6,6,1,6,
%T A033773 8,6,5,8,4,1,9,6,6,8,8,6,2,6,8,13,8,10,10,0,12,14,8,8
%N A033773 Product t2(q^d); d | 14, where t2 = theta2(q)/(2*q^(1/4)).
%Y A033773 Adjacent sequences: A033770 A033771 A033772 this_sequence A033774 A033775 A033776
%Y A033773 Sequence in context: A035392 A007149 A028832 this_sequence A029275 A058739 A128627
%K A033773 nonn
%O A033773 0,4
%A A033773 njas

%I A029275
%S A029275 1,0,0,1,0,1,2,0,1,2,2,2,3,2,2,5,4,3,6,4,6,8,6,7,9,9,10,
%T A029275 11,11,11,16,14,14,18,16,19,23,19,22,25,26,27,30,29,30,
%U A029275 37,36,35,42,39,44,49,46,49,54,55,58,61,62,63,73,71,72
%N A029275 Expansion of 1/((1-x^3)(1-x^5)(1-x^6)(1-x^10)).
%Y A029275 Adjacent sequences: A029272 A029273 A029274 this_sequence A029276 A029277 A029278
%Y A029275 Sequence in context: A007149 A028832 A033773 this_sequence A058739 A128627 A105422
%K A029275 nonn
%O A029275 0,7
%A A029275 njas

%I A058739
%S A058739 1,0,2,0,1,2,2,2,4,2,5,6,7,6,12,8,13,14,19,16,25,20,31,32,40
%N A058739 McKay-Thompson series of class 66A for Monster.
%D A058739 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%e A058739 T66A = 1/q + 2*q + q^3 + 2*q^4 + 2*q^5 + 2*q^6 + 4*q^7 + 2*q^8 + 5*q^9 + ...
%Y A058739 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%Y A058739 Adjacent sequences: A058736 A058737 A058738 this_sequence A058740 A058741 A058742
%Y A058739 Sequence in context: A028832 A033773 A029275 this_sequence A128627 A105422 A128584
%K A058739 nonn
%O A058739 -1,3
%A A058739 njas, Nov 27, 2000

%I A128627
%S A128627 1,0,1,1,0,1,1,2,0,1,2,2,3,0,1,2,5,3,4,0,1,4,6,9,4,5,0,1,4,13,12,14,5,6,
%T A128627 0,1,7,16,28,20,20,6,7,0,1
%N A128627 Triangular array illustrating the application of cyclic partitions to the computation of partitions of an integer into parts of k kinds. (cf A060850).
%C A128627 The array is constructed by summing sequences associated with each cyclic partition as indicated below: (n' here denotes the sum of preceding sequences).
%C A128627 4.......1......2......3......................................
%C A128627 22......1......3......6......................................
%C A128627 4'......2......5......9......................................
%C A128627 5.......1......2......3......4...............................
%C A128627 32......1......4......9......16..............................
%C A128627 5'......2......6......12......20.............................
%C A128627 6.......1......2......3.......4.......5.......6.......7.......8......9
%C A128627 42......1......4......9......16......25......36......49......64......81
%C A128627 33......1......3......6......10......15......21......28......36......45
%C A128627 222.....1......4......10......20......35......56......84......120...165
%C A128627 6'......4......13.....28......50......80....119......168......228...300
%C A128627 7.......1......2......3.......4.......5.......6.......7.......8.......9
%C A128627 52......1......4......9......16......25......36......49......64......81
%C A128627 43......1......4......9......16......25......36......49......64......81
%C A128627 322.....1......6......18......40.....75.....126......196.....288.....405
%C A128627 7'......4......16.....39......76......130...204......301.....424.....576
%C A128627 8.......1......2......3.......4.......5.......6.......7.......8.......9
%C A128627 62......1......4......9......16......25......36......49......64......81
%C A128627 53......1......4......9......16......25......36......49......64......81
%C A128627 44......1......3......6......10......15......21......28......36......45
%C A128627 422.....1......6......18......40......75......126......196......288......405
%C A128627 332.....1......6......18......40......75......126......196......288......405
%C A128627 2222....1......5......15......35......70......126......210......330......495
%C A128627 8'......7......30.....78......161......290......477......735......1078......1521
%e A128627 The diagonal 9th diagonal of A060850 is 22 185 810 2580 6765 ... and can be computed from a(n) and A007318 as illustrated:
%e A128627 1..................
%e A128627 0......1............
%e A128627 1......0......1......
%e A128627 1......2......0......1
%e A128627 2......2......3......0
%e A128627 2......5......3......4
%e A128627 4......6......9......4
%e A128627 4......13....12......14
%e A128627 7......16....28......20
%e A128627 .......30....39......50
%e A128627 .............78......76
%e A128627 ....................161
%e A128627 times
%e A128627 1..................
%e A128627 1......9............
%e A128627 1......8......45......
%e A128627 1......7......36......165
%e A128627 1......6......28......120
%e A128627 1......5......21......84
%e A128627 1......4......15......56
%e A128627 1......3......10......35
%e A128627 1......2......6......20
%e A128627 .......1......3......10
%e A128627 ..............1......4
%e A128627 .....................1
%e A128627 yields
%e A128627 1..................
%e A128627 0......9............
%e A128627 1......0......45......
%e A128627 1......14......0.......165
%e A128627 2......12......84......0
%e A128627 2......25......63......336
%e A128627 4......24......135.....224
%e A128627 4......39......120.....490
%e A128627 7......32......168.....400
%e A128627 .......30......117.....500
%e A128627 ...............78......304
%e A128627 .......................161
%e A128627 summing to
%e A128627 22.....185.....810.....2580 ...
%Y A128627 Cf. A060850 A007318 A002865.
%Y A128627 Adjacent sequences: A128624 A128625 A128626 this_sequence A128628 A128629 A128630
%Y A128627 Sequence in context: A033773 A029275 A058739 this_sequence A105422 A128584 A080099
%K A128627 nonn,tabl
%O A128627 1,8
%A A128627 Alford Arnold (Alford1940(AT)aol.com), Mar 22 2007

%I A105422
%S A105422 1,0,1,1,0,1,1,2,0,1,2,2,3,0,1,3,5,3,4,0,1,5,8,9,4,5,0,1,8,15,15,14,5,6,
%T A105422 0,1,13,26,31,24,20,6,7,0,1,21,46,57,54,35,27,7,8,0,1,34,80,108,104,85,
%U A105422 48,35,8,9,0,1,55,139,199,209,170,125,63,44,9,10,0,1,89,240,366,404,360
%N A105422 Triangle read by rows: T(n,k) is the number of compositions of n having exactly k parts equal to 1.
%F A105422 G.f.=(1-z)/(1-z-z^2-tz+tz^2).
%e A105422 T(6,2)=9 because we have (1,1,4),(1,4,1),(4,1,1),(1,1,2,2),(1,2,1,2),(1,2,2,1),(2,1,1,2),(2,1,2,1) and (2,2,1,1).
%e A105422 Triangle begins:
%e A105422 1;
%e A105422 0,1;
%e A105422 1,0,1;
%e A105422 1,2,0,1;
%e A105422 2,2,3,0,1;
%p A105422 G:=(1-z)/(1-z-z^2-t*z+t*z^2): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 14 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 13 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form
%Y A105422 Column 0 yields A000045 (the Fibonacci numbers). Column 1 yields A006367. Column 2 yields A105423. Row sums yield A011782.
%Y A105422 Adjacent sequences: A105419 A105420 A105421 this_sequence A105423 A105424 A105425
%Y A105422 Sequence in context: A029275 A058739 A128627 this_sequence A128584 A080099 A127711
%K A105422 nonn,tabl
%O A105422 0,8
%A A105422 Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 07 2005

%I A128584
%S A128584 1,1,0,0,2,0,1,2,3,0,0,0,0,4,0,1,0,0,4,5,0,0,0,0,0,0,6,1,2,3,4,5,6,7,0,
%T A128584 0,2,3,0,5,6,0,8,0,1,2,3,0,0,6,7,8,9,0,1,2,3,0,0,6,7,8,9,0
%V A128584 1,1,0,0,2,0,1,-2,3,0,0,0,0,4,0,1,0,0,-4,5,0,0,0,0,0,0,6,1,-2,3,-4,5,-6,7,0,0,2,-3,0,
%W A128584 -5,-6,0,8,0,1,-2,3,0,0,-6,7,-8,9,0,1,-2,3,0,0,-6,7,-8,9,0
%N A128584 A054521 * A128064 (unsigned).
%C A128584 Row sums = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6, 4, 6, 4,...)
%F A128584 A054521 * A128064 (unsigned) as infinite lower triangular matrices.
%e A128584 First few rows of the triangle are:
%e A128584 1;
%e A128584 1, 0;
%e A128584 0, 2, 0;
%e A128584 1, -2, 3, 0;
%e A128584 0, 0, 0, 4, 0;
%e A128584 1, 0, 0, -4, 5, 0;
%e A128584 ...
%Y A128584 Cf. A054521, A128064, A000010.
%Y A128584 Adjacent sequences: A128581 A128582 A128583 this_sequence A128585 A128586 A128587
%Y A128584 Sequence in context: A058739 A128627 A105422 this_sequence A080099 A127711 A069584
%K A128584 nonn,tabl
%O A128584 1,5
%A A128584 Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 11 2007

%I A080099
%S A080099 0,0,1,0,0,2,0,1,2,3,0,0,0,0,4,0,1,0,1,4,5,0,0,2,2,4,4,6,0,1,2,3,4,5,6,
%T A080099 7,0,0,0,0,0,0,0,0,8,0,1,0,1,0,1,0,1,8,9,0,0,2,2,0,0,2,2,8,8,10,0,1,2,3,
%U A080099 0,1,2,3,8,9,10,11,0,0,0,0,4,4,4,4,8,8,8,8,12,0,1,0,1,4,5,4,5,8,9,8,9
%N A080099 Triangle T(n,k) = n AND k, 0<=k<=n, bitwise logical AND, read by rows.
%C A080099 A080100(n) = number of numbers k such that n AND k = 0 in n-th row of the triangular array.
%H A080099 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AND.html">AND.</a>
%Y A080099 Cf. A080098, A051933.
%Y A080099 Adjacent sequences: A080096 A080097 A080098 this_sequence A080100 A080101 A080102
%Y A080099 Sequence in context: A128627 A105422 A128584 this_sequence A127711 A069584 A014587
%K A080099 nonn,tabl
%O A080099 0,6
%A A080099 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jan 28 2003

%I A127711
%S A127711 1,0,1,0,1,2,0,1,2,3,0,0,0,3,5,0,1,2,0,5,8,0,0,0,0,0,8,13,0,1,2,3,0,0,
%T A127711 13,21,0,0,0,0,0,0,0,21,34,0,0,0,3,5,0,0,0,34,55,0,0,0,0,0,0,0,0,0,55,
%U A127711 89,0,1,2,0,5,8,0,0,0
%V A127711 1,0,1,0,-1,2,0,1,-2,3,0,0,0,-3,5,0,-1,2,0,-5,8,0,0,0,0,0,-8,13,0,1,-2,3,0,0,-13,21,0,
%W A127711 0,0,0,0,0,0,-21,34,0,0,0,-3,5,0,0,0,-34,55,0,0,0,0,0,0,0,0,0,-55,89,0,-1,2,0,-5,8,0,0,
%X A127711 0
%N A127711 Inverse of the triangle A(n,k)=if(k<=n,if(n<=2k,1/F(n+1),0),0).
%C A127711 It is conjectured that all elements of this triangle are integers. Row sums are A127712.
%e A127711 Triangle begins
%e A127711 1,
%e A127711 0, 1,
%e A127711 0, -1, 2,
%e A127711 0, 1, -2, 3,
%e A127711 0, 0, 0, -3, 5,
%e A127711 0, -1, 2, 0, -5, 8,
%e A127711 0, 0, 0, 0, 0, -8, 13,
%e A127711 0, 1, -2, 3, 0, 0, -13, 21,
%e A127711 0, 0, 0, 0, 0, 0, 0, -21, 34,
%e A127711 0, 0, 0, -3, 5, 0, 0, 0, -34, 55,
%e A127711 0, 0, 0, 0, 0, 0, 0, 0, 0, -55, 89,
%e A127711 0, -1, 2, 0, -5, 8, 0, 0, 0, 0, -89, 144,
%e A127711 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -144, 233
%e A127711 Triangle is the inverse of the triangle
%e A127711 1,
%e A127711 0, 1,
%e A127711 0, 1/2, 1/2,
%e A127711 0, 0, 1/3, 1/3,
%e A127711 0, 0, 1/5, 1/5, 1/5,
%e A127711 0, 0, 0, 1/8, 1/8, 1/8,
%e A127711 0, 0, 0, 1/13, 1/13, 1/13, 1/13,
%e A127711 0, 0, 0, 0, 1/21, 1/21, 1/21, 1/21,
%e A127711 0, 0, 0, 0, 1/34, 1/34, 1/34, 1/34, 1/34
%Y A127711 Adjacent sequences: A127708 A127709 A127710 this_sequence A127712 A127713 A127714
%Y A127711 Sequence in context: A105422 A128584 A080099 this_sequence A069584 A014587 A025658
%K A127711 sign,tabl
%O A127711 0,6
%A A127711 Paul Barry (pbarry(AT)wit.ie), Jan 24 2007

%I A069584
%S A069584 0,1,2,0,1,2,3,0,0,1,2,3,4,5,6,0,1,2,3,4,5,6,7,8,0,1,0,1,2,3,4,0,1,2,3,
%T A069584 0,1,2,3,4,5,6,7,8,9,10,11,12,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,0,1,2,
%U A069584 3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14
%N A069584 a(n) = n - largest perfect power <= n.
%C A069584 a(n) = 0 if n = m^p that is if n is a full power (square, cube etc).
%C A069584 As Catalan's conjecture is now proved, n=8=2^3, n+1=9=3^2 is the only solution for a(n+1) = a(n) = 0.
%H A069584 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/news/2002-05-05/catalan/">Catalan's conjecture</a>
%Y A069584 Cf. A001597, a(n)=n-A081676(n).
%Y A069584 Adjacent sequences: A069581 A069582 A069583 this_sequence A069585 A069586 A069587
%Y A069584 Sequence in context: A128584 A080099 A127711 this_sequence A014587 A025658 A025673
%K A069584 easy,nonn
%O A069584 1,3
%A A069584 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 24 2002
%E A069584 Edited by Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 26 2003

%I A014587
%S A014587 0,1,2,0,1,2,3,0,1,2,0,1,2,3,0,1,2,0,1,2,3,0,1,2,3,0,1,2,0,1,2,3,0,
%T A014587 1,2,0,1,2,3,0,1,2,0,1,2,3,0,1,2,3,0,1,2,0,1,2,3,0,1,2,0,1,2,3,0,1,
%U A014587 2,0,1,2,3,0,1,2,3,0,1,2,0,1,2,3,0,1,2,0,1,2,3,0,1,2,0,1,2,3,0,1,2
%N A014587 Nim function for Take-a-Factorial-Game.
%D A014587 R. K. Guy, Unsolved Problems in Number Theory, E26.
%H A014587 Achim Flammenkamp, <a href="http://www.uni-bielefeld.de/~achim/diss.ps.gz">Lange Perioden in Subtraktions-Spielen</a>, Dissertation, Dept. Math., University of Bielefeld, Germany.
%Y A014587 Adjacent sequences: A014584 A014585 A014586 this_sequence A014588 A014589 A014590
%Y A014587 Sequence in context: A080099 A127711 A069584 this_sequence A025658 A025673 A025688
%K A014587 nonn
%O A014587 0,3
%A A014587 Achim Flammenkamp (achim(AT)HRZ.Uni-Bielefeld.DE)

%I A025658
%S A025658 0,0,1,0,1,2,0,1,2,3,0,1,2,3,0,4,1,2,3,0,4,1,5,2,3,0,4,1,5,2,6,3,0,4,1,5,
%T A025658 2,6,3,0,7,4,1,5,2,6,3,0,7,4,1,8,5,2,6,3,0,7,4,1,8,5,2,9,6,3,0,7,4,1,8,5,
%U A025658 2,9,6,3,10,0,7,4,1,8,5,2,9,6,3,10,0,7,4,11,1,8,5,2,9,6,3,10,0,7,4,11,1
%N A025658 Exponent of 6 (value of j) in n-th number of form 4^i*6^j.
%Y A025658 Differs from A025673 at a(2805).
%Y A025658 Adjacent sequences: A025655 A025656 A025657 this_sequence A025659 A025660 A025661
%Y A025658 Sequence in context: A127711 A069584 A014587 this_sequence A025673 A025688 A025645
%K A025658 nonn
%O A025658 1,6
%A A025658 David W. Wilson (davidwwilson(AT)comcast.net)

%I A025673
%S A025673 0,0,1,0,1,2,0,1,2,3,0,1,2,3,0,4,1,2,3,0,4,1,5,2,3,0,4,1,5,2,6,3,0,4,1,5,
%T A025673 2,6,3,0,7,4,1,5,2,6,3,0,7,4,1,8,5,2,6,3,0,7,4,1,8,5,2,9,6,3,0,7,4,1,8,5,
%U A025673 2,9,6,3,10,0,7,4,1,8,5,2,9,6,3,10,0,7,4,11,1,8,5,2,9,6,3,10,0,7,4,11,1
%N A025673 Exponent of 8 (value of j) in n-th number of form 5^i*8^j.
%Y A025673 Differs from A025658 at a(2805).
%Y A025673 Adjacent sequences: A025670 A025671 A025672 this_sequence A025674 A025675 A025676
%Y A025673 Sequence in context: A069584 A014587 A025658 this_sequence A025688 A025645 A025680
%K A025673 nonn
%O A025673 1,6
%A A025673 David W. Wilson (davidwwilson(AT)comcast.net)

%I A025688
%S A025688 0,0,1,0,1,2,0,1,2,3,0,1,2,3,0,4,1,2,3,0,4,1,5,2,3,0,4,1,5,2,6,3,0,4,1,5,
%T A025688 2,6,3,7,0,4,1,5,2,6,3,7,0,4,8,1,5,2,6,3,7,0,4,8,1,5,9,2,6,3,7,0,4,8,1,5,
%U A025688 9,2,6,10,3,7,0,4,8,1,5,9,2,6,10,3,7,0,11,4,8,1,5,9,2,6,10,3,7,0,11,4,8
%N A025688 Exponent of 10 (value of j) in n-th number of form 6^i*10^j.
%Y A025688 Adjacent sequences: A025685 A025686 A025687 this_sequence A025689 A025690 A025691
%Y A025688 Sequence in context: A014587 A025658 A025673 this_sequence A025645 A025680 A025667
%K A025688 nonn
%O A025688 1,6
%A A025688 David W. Wilson (davidwwilson(AT)comcast.net)

%I A025645
%S A025645 0,0,1,0,1,2,0,1,2,3,0,1,2,3,0,4,1,2,3,0,4,1,5,2,3,0,4,1,5,2,6,3,0,4,1,5,
%T A025645 2,6,3,7,0,4,1,5,2,6,3,7,0,4,8,1,5,2,6,3,7,0,4,8,1,5,9,2,6,3,7,0,4,8,1,5,
%U A025645 9,2,6,10,3,7,0,4,8,1,5,9,2,6,10,3,7,11,0,4,8,1,5,9,2,6,10,3,7,11,0,4,8
%N A025645 Exponent of 4 (value of j) in n-th number of form 3^i*4^j.
%Y A025645 Adjacent sequences: A025642 A025643 A025644 this_sequence A025646 A025647 A025648
%Y A025645 Sequence in context: A025658 A025673 A025688 this_sequence A025680 A025667 A025689
%K A025645 nonn
%O A025645 1,6
%A A025645 David W. Wilson (davidwwilson(AT)comcast.net)

%I A025680
%S A025680 0,0,1,0,1,2,0,1,2,3,0,1,2,3,4,0,1,2,3,4,0,5,1,2,3,4,0,5,1,6,2,3,4,0,5,1,
%T A025680 6,2,7,3,4,0,5,1,6,2,7,3,8,4,0,5,1,6,2,7,3,8,4,0,9,5,1,6,2,7,3,8,4,0,9,5,
%U A025680 1,10,6,2,7,3,8,4,0,9,5,1,10,6,2,11,7,3,8,4,0,9,5,1,10,6,2,11,7,3,12,8,4
%N A025680 Exponent of 9 (value of j) in n-th number of form 6^i*9^j.
%Y A025680 Adjacent sequences: A025677 A025678 A025679 this_sequence A025681 A025682 A025683
%Y A025680 Sequence in context: A025673 A025688 A025645 this_sequence A025667 A025689 A025650
%K A025680 nonn
%O A025680 1,6
%A A025680 David W. Wilson (davidwwilson(AT)comcast.net)

%I A025667
%S A025667 0,0,1,0,1,2,0,1,2,3,0,1,2,3,4,0,1,2,3,4,0,5,1,2,3,4,0,5,1,6,2,3,4,0,5,1,
%T A025667 6,2,7,3,4,0,5,1,6,2,7,3,8,4,0,5,1,6,2,7,3,8,4,9,0,5,1,6,2,7,3,8,4,9,0,5,
%U A025667 10,1,6,2,7,3,8,4,9,0,5,10,1,6,11,2,7,3,8,4,9,0,5,10,1,6,11,2,7,12,3,8,4
%N A025667 Exponent of 7 (value of j) in n-th number of form 5^i*7^j.
%Y A025667 Adjacent sequences: A025664 A025665 A025666 this_sequence A025668 A025669 A025670
%Y A025667 Sequence in context: A025688 A025645 A025680 this_sequence A025689 A025650 A025674
%K A025667 nonn
%O A025667 1,6
%A A025667 David W. Wilson (davidwwilson(AT)comcast.net)

%I A025689
%S A025689 0,0,1,0,1,2,0,1,2,3,0,1,2,3,4,0,1,2,3,4,5,0,1,2,3,4,5,0,6,1,2,3,4,5,0,6,
%T A025689 1,7,2,3,4,5,0,6,1,7,2,8,3,4,5,0,6,1,7,2,8,3,9,4,5,0,6,1,7,2,8,3,9,4,10,
%U A025689 5,0,6,1,7,2,8,3,9,4,10,5,0,11,6,1,7,2,8,3,9,4,10,5,0,11,6,1,12,7,2,8,3
%N A025689 Exponent of 10 (value of j) in n-th number of form 7^i*10^j.
%Y A025689 Adjacent sequences: A025686 A025687 A025688 this_sequence A025690 A025691 A025692
%Y A025689 Sequence in context: A025645 A025680 A025667 this_sequence A025650 A025674 A025681
%K A025689 nonn
%O A025689 1,6
%A A025689 David W. Wilson (davidwwilson(AT)comcast.net)

%I A025650
%S A025650 0,0,1,0,1,2,0,1,2,3,0,1,2,3,4,0,1,2,3,4,5,0,1,2,3,4,5,6,0,1,2,3,4,5,6,0,
%T A025650 7,1,2,3,4,5,6,0,7,1,8,2,3,4,5,6,0,7,1,8,2,9,3,4,5,6,0,7,1,8,2,9,3,10,4,
%U A025650 5,6,0,7,1,8,2,9,3,10,4,11,5,6,0,7,1,8,2,9,3,10,4,11,5,12,6,0,7,1,8,2,9
%N A025650 Exponent of 5 (value of j) in n-th number of form 4^i*5^j.
%Y A025650 Differs from A025674 at a(1881).
%Y A025650 Adjacent sequences: A025647 A025648 A025649 this_sequence A025651 A025652 A025653
%Y A025650 Sequence in context: A025680 A025667 A025689 this_sequence A025674 A025681 A025659
%K A025650 nonn
%O A025650 1,6
%A A025650 David W. Wilson (davidwwilson(AT)comcast.net)

%I A025674
%S A025674 0,0,1,0,1,2,0,1,2,3,0,1,2,3,4,0,1,2,3,4,5,0,1,2,3,4,5,6,0,1,2,3,4,5,6,0,
%T A025674 7,1,2,3,4,5,6,0,7,1,8,2,3,4,5,6,0,7,1,8,2,9,3,4,5,6,0,7,1,8,2,9,3,10,4,
%U A025674 5,6,0,7,1,8,2,9,3,10,4,11,5,6,0,7,1,8,2,9,3,10,4,11,5,12,6,0,7,1,8,2,9
%N A025674 Exponent of 8 (value of j) in n-th number of form 6^i*8^j.
%Y A025674 Differs from A025650 at a(1881).
%Y A025674 Adjacent sequences: A025671 A025672 A025673 this_sequence A025675 A025676 A025677
%Y A025674 Sequence in context: A025667 A025689 A025650 this_sequence A025681 A025659 A025690
%K A025674 nonn
%O A025674 1,6
%A A025674 David W. Wilson (davidwwilson(AT)comcast.net)

%I A025681
%S A025681 0,0,1,0,1,2,0,1,2,3,0,1,2,3,4,0,1,2,3,4,5,0,1,2,3,4,5,6,0,1,2,3,4,5,6,7,
%T A025681 0,1,2,3,4,5,6,7,0,8,1,2,3,4,5,6,7,0,8,1,9,2,3,4,5,6,7,0,8,1,9,2,10,3,4,
%U A025681 5,6,7,0,8,1,9,2,10,3,11,4,5,6,7,0,8,1,9,2,10,3,11,4,12,5,6,7,0,8,1,9,2
%N A025681 Exponent of 9 (value of j) in n-th number of form 7^i*9^j.
%Y A025681 Adjacent sequences: A025678 A025679 A025680 this_sequence A025682 A025683 A025684
%Y A025681 Sequence in context: A025689 A025650 A025674 this_sequence A025659 A025690 A025668
%K A025681 nonn
%O A025681 1,6
%A A025681 David W. Wilson (davidwwilson(AT)comcast.net)

%I A025659
%S A025659 0,0,1,0,1,2,0,1,2,3,0,1,2,3,4,0,1,2,3,4,5,0,1,2,3,4,5,6,0,1,2,3,4,5,6,7,
%T A025659 0,1,2,3,4,5,6,7,8,0,1,2,3,4,5,6,7,8,0,9,1,2,3,4,5,6,7,8,0,9,1,10,2,3,4,
%U A025659 5,6,7,8,0,9,1,10,2,11,3,4,5,6,7,8,0,9,1,10,2,11,3,12,4,5,6,7,8,0,9,1,10
%N A025659 Exponent of 6 (value of j) in n-th number of form 5^i*6^j.
%Y A025659 Adjacent sequences: A025656 A025657 A025658 this_sequence A025660 A025661 A025662
%Y A025659 Sequence in context: A025650 A025674 A025681 this_sequence A025690 A025668 A048151
%K A025659 nonn
%O A025659 1,6
%A A025659 David W. Wilson (davidwwilson(AT)comcast.net)

%I A025690
%S A025690 0,0,1,0,1,2,0,1,2,3,0,1,2,3,4,0,1,2,3,4,5,0,1,2,3,4,5,6,0,1,2,3,4,5,6,7,
%T A025690 0,1,2,3,4,5,6,7,8,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,10,1,2,3,4,
%U A025690 5,6,7,8,9,0,10,1,11,2,3,4,5,6,7,8,9,0,10,1,11,2,12,3,4,5,6,7,8,9,0,10,1
%N A025690 Exponent of 10 (value of j) in n-th number of form 8^i*10^j.
%Y A025690 Adjacent sequences: A025687 A025688 A025689 this_sequence A025691 A025692 A025693
%Y A025690 Sequence in context: A025674 A025681 A025659 this_sequence A025668 A048151 A002262
%K A025690 nonn
%O A025690 1,6
%A A025690 David W. Wilson (davidwwilson(AT)comcast.net)

%I A025668
%S A025668 0,0,1,0,1,2,0,1,2,3,0,1,2,3,4,0,1,2,3,4,5,0,1,2,3,4,5,6,0,1,2,3,4,5,6,7,
%T A025668 0,1,2,3,4,5,6,7,8,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,10,0,1,2,3,4,
%U A025668 5,6,7,8,9,10,11,0,1,2,3,4,5,6,7,8,9,10,11,0,12,1,2,3,4,5,6,7,8,9,10,11
%N A025668 Exponent of 7 (value of j) in n-th number of form 6^i*7^j.
%Y A025668 Adjacent sequences: A025665 A025666 A025667 this_sequence A025669 A025670 A025671
%Y A025668 Sequence in context: A025681 A025659 A025690 this_sequence A048151 A002262 A025675
%K A025668 nonn
%O A025668 1,6
%A A025668 David W. Wilson (davidwwilson(AT)comcast.net)

%I A048151
%S A048151 0,1,0,1,2,0,1,2,3,0,1,2,3,4,0,1,2,3,4,5,0,1,2,3,4,5,6,0,1,2,3,4,5,6,7,
%T A048151 0,1,2,3,4,5,6,7,8,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,10,0,1,2,3,
%U A048151 4,5,6,7,8,9,10,11,0,1,2,3,4,5,6,7,8,9,10,11,12
%N A048151 Triangular array T read by rows: T(n,k)=k mod n, for k=1,2,...,n, n=1,2,...
%e A048151 Rows: {0}; {1,0}; {1,2,0}; ...
%Y A048151 As a sequence, essentially same as A002262.
%Y A048151 Adjacent sequences: A048148 A048149 A048150 this_sequence A048152 A048153 A048154
%Y A048151 Sequence in context: A025659 A025690 A025668 this_sequence A002262 A025675 A025682
%K A048151 nonn,tabl
%O A048151 1,5
%A A048151 Clark Kimberling (ck6(AT)evansville.edu)

%I A002262
%S A002262 0,0,1,0,1,2,0,1,2,3,0,1,2,3,4,0,1,2,3,4,5,0,1,2,3,4,5,6,0,1,2,3,4,
%T A002262 5,6,7,0,1,2,3,4,5,6,7,8,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,10,
%U A002262 0,1,2,3,4,5,6,7,8,9,10,11,0,1,2,3,4,5,6,7,8,9,10,11,12,0,1,2,3,4,5
%N A002262 Integers 0 to n followed by integers 0 to n+1 etc.
%C A002262 a(n) = n - the largest triangular number <= n. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Dec 25 2001
%C A002262 The PARI functions t1, t2 can be used to read a square array T(n,k) (n >= 0, k >= 0) by antidiagonals downwards: n -> T(t1(n), t2(n)). - Michael Somos, Aug 23, 2002
%C A002262 Values x of unique solution pair (x,y) to equation T(x+y) + x = n, where T(k)=A000217(k). - Lekraj Beedassy (blekraj(AT)yahoo.com), Aug 21 2004
%H A002262 M. Somos, <a href="http://www.research.att.com/~njas/sequences/a073189.txt">Sequences used for indexing triangular or square arrays</a>
%F A002262 a(n) = (n-((trinv(n)*(trinv(n)-1))/2)); trinv := n -> floor((1+sqrt(1+8*n))/2); # Gives integral inverses of triangular numbers.
%F A002262 a(n)=n-A000217(A003056(n))=n-A057944(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Aug 21 2004
%p A002262 A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);
%o A002262 (PARI) a(n)=n-binomial(floor(1/2+sqrt(2+2*n)),2)
%o A002262 (PARI) t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)),2) /* A002262 */
%o A002262 (PARI) t2(n)=binomial(floor(3/2+sqrt(2+2*n)),2)-(n+1) /* A025581 */
%Y A002262 A002260(n)=1+a(n)
%Y A002262 Cf. A025675, A025682, A025691, A002024, A048645, A004736, A025581. As a sequence, essentially same as A048151.
%Y A002262 Adjacent sequences: A002259 A002260 A002261 this_sequence A002263 A002264 A002265
%Y A002262 Sequence in context: A025690 A025668 A048151 this_sequence A025675 A025682 A025691
%K A002262 nonn,tabl,easy,nice
%O A002262 0,6
%A A002262 Angele Hamel (amh(AT)maths.soton.ac.uk)

%I A025675
%S A025675 0,0,1,0,1,2,0,1,2,3,0,1,2,3,4,0,1,2,3,4,5,0,1,2,3,4,5,6,0,1,2,3,4,5,6,7,
%T A025675 0,1,2,3,4,5,6,7,8,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,10,0,1,2,3,4,
%U A025675 5,6,7,8,9,10,11,0,1,2,3,4,5,6,7,8,9,10,11,12,0,1,2,3,4,5,6,7,8,9,10,11
%N A025675 Exponent of 8 (value of j) in n-th number of form 7^i*8^j.
%Y A025675 Differs from A002262 at a(136).
%Y A025675 Adjacent sequences: A025672 A025673 A025674 this_sequence A025676 A025677 A025678
%Y A025675 Sequence in context: A025668 A048151 A002262 this_sequence A025682 A025691 A099173
%K A025675 nonn
%O A025675 1,6
%A A025675 David W. Wilson (davidwwilson(AT)comcast.net)

%I A025682
%S A025682 0,0,1,0,1,2,0,1,2,3,0,1,2,3,4,0,1,2,3,4,5,0,1,2,3,4,5,6,0,1,2,3,4,5,6,7,
%T A025682 0,1,2,3,4,5,6,7,8,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,10,0,1,2,3,4,
%U A025682 5,6,7,8,9,10,11,0,1,2,3,4,5,6,7,8,9,10,11,12,0,1,2,3,4,5,6,7,8,9,10,11
%N A025682 Exponent of 9 (value of j) in n-th number of form 8^i*9^j.
%Y A025682 Differs from A002262 at a(190).
%Y A025682 Adjacent sequences: A025679 A025680 A025681 this_sequence A025683 A025684 A025685
%Y A025682 Sequence in context: A048151 A002262 A025675 this_sequence A025691 A099173 A108456
%K A025682 nonn
%O A025682 1,6
%A A025682 David W. Wilson (davidwwilson(AT)comcast.net)

%I A025691
%S A025691 0,0,1,0,1,2,0,1,2,3,0,1,2,3,4,0,1,2,3,4,5,0,1,2,3,4,5,6,0,1,2,3,4,5,6,7,
%T A025691 0,1,2,3,4,5,6,7,8,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,10,0,1,2,3,4,
%U A025691 5,6,7,8,9,10,11,0,1,2,3,4,5,6,7,8,9,10,11,12,0,1,2,3,4,5,6,7,8,9,10,11
%N A025691 Exponent of 10 (value of j) in n-th number of form 9^i*10^j.
%Y A025691 Differs from A002262 at a(253).
%Y A025691 Adjacent sequences: A025688 A025689 A025690 this_sequence A025692 A025693 A025694
%Y A025691 Sequence in context: A002262 A025675 A025682 this_sequence A099173 A108456 A089107
%K A025691 nonn
%O A025691 1,6
%A A025691 David W. Wilson (davidwwilson(AT)comcast.net)

%I A099173
%S A099173 0,0,1,0,1,2,0,1,2,3,0,1,2,4,4,0,1,2,5,8,5,0,1,2,6,12,16,6,0,1,2,7,16,
%T A099173 29,32,7,0,1,2,8,20,44,70,64,8,0,1,2,9,24,61,120,169,128,9,0,1,2,10,28,
%U A099173 80,182,328,408,256,10,0,1,2,11,32,101,256,547,896,985,512,11,0,1
%N A099173 Array T(k,n) read by diagonals: g.f. of k-th row x/(1-2x-(k-1)x^2).
%H A099173 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0409509">Prove or disprove. 100 Conjectures from the OEIS</a>, #16.
%F A099173 T(k, n) = Sum{i=0..[n/2], k^i * C(n, 2i+1) }.
%F A099173 Recurrence: T(k, 0)=0, T(k, 1)=1, T(k, n) = 2T(k, n-1) + (k-1)T(k, n-2).
%e A099173 0,1,2,3,4,5,6,
%e A099173 0,1,2,4,8,16,32,
%e A099173 0,1,2,5,12,29,70,
%e A099173 0,1,2,6,16,44,120,
%e A099173 0,1,2,7,20,61,182,
%e A099173 0,1,2,8,24,80,256,
%o A099173 (PARI) T(k,n)=sum(i=0,n\2,k^i*binomial(n,2*i+1))
%Y A099173 Rows 0-12 are A001477, A000079, A000129, A002605, A015518, A063727, A002532, A083099, A015519, A003683, A002534, A083102, A015520, A091914.
%Y A099173 Columns 0-4 are A000004, A000012, A009056, A008586.
%Y A099173 Adjacent sequences: A099170 A099171 A099172 this_sequence A099174 A099175 A099176
%Y A099173 Sequence in context: A025675 A025682 A025691 this_sequence A108456 A089107 A117398
%K A099173 nonn
%O A099173 0,6
%A A099173 Ralf Stephan, Oct 13 2004

%I A108456
%S A108456 1,0,1,0,1,2,0,1,2,3,0,1,3,4,5,0,1,3,6,7,7,0,1,4,8,12,12,11,0,1,4,10,16,
%T A108456 21,19,15,0,1,5,12,23,31,36,30,22,0,1,5,15,28,45,55,58,45,30,0,1,6,17,
%U A108456 37,60,84,94,92,67,42,0,1,6,20,44,80,115,147,153,140,97,56,0,1,7,23
%N A108456 Table read by antidiagonals: T(n,k) = number of partitions of (n,k) into pairs (i,j) with i>0, j>=0.
%C A108456 (a,b)+(x,y)=(a+x,b+y); unit is (0,0).
%H A108456 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%F A108456 Euler transform of table whose g.f. is x/((1-x)*(1-y)).
%e A108456 1 0 0 0 0 ...
%e A108456 1 1 1 1 1 ...
%e A108456 2 2 3 3 4 ...
%e A108456 3 4 6 8 10 ...
%e A108456 5 7 12 16 23 ...
%e A108456 (3,2)=(2,2)+(1,0)=(2,1)+(1,1)=(2,0)+(1,2)=(1,2)+(1,0)+(1,0)=(1,1)+(1,1)+(1,0), so a(3,2)=6.
%Y A108456 Cf. A108461, A108455. Columns 0-1: A000041, A000070. Main diagonal: A108457.
%Y A108456 Adjacent sequences: A108453 A108454 A108455 this_sequence A108457 A108458 A108459
%Y A108456 Sequence in context: A025682 A025691 A099173 this_sequence A089107 A117398 A071486
%K A108456 nonn,tabl
%O A108456 0,6
%A A108456 Christian G. Bower (bowerc(AT)usa.net), Jun 03 2005

%I A089107
%S A089107 1,0,1,0,1,2,0,1,2,3,0,1,3,5,5,0,1,3,7,9,8,0,1,4,10,17,19,13,0,1,4,13,
%T A089107 25,37,34,21,0,1,5,16,38,64,77,65,34,0,1,5,20,51,102,146,158,115,55,0,1,
%U A089107 6,24,70,154,259,331,314,210,89,0,1,6,28,89,222,418,626,710,611,368,144
%N A089107 Square array T(r,j) (r>=1, j>=1) read by antidiagonals, where T(r,j) is the convoluted convolved Fibonacci number G_j^(r) (see the Moree paper).
%H A089107 P. Moree, <a href="http://arXiv.org/abs/math.CO/0311205">Convoluted convolved Fibonacci numbers</a>
%e A089107 Triangle begins:
%e A089107 1
%e A089107 0 1
%e A089107 0 1 2
%e A089107 0 1 2 3
%e A089107 0 1 3 5 5
%e A089107 Array begins:
%e A089107 [1, 1, 2, 3, 5, 8, 13, 21, ...],
%e A089107 [0, 1, 2, 5, 9, 19, 34, 65, ...],
%e A089107 [0, 1, 3, 7, 17, 37, 77, 158, ...],
%e A089107 [0, 1, 3, 10, 25, 64, 146, 331, ...],
%e A089107 [0, 1, 4, 13, 38, 102, 259, 626, ...],
%e A089107 [0, 1, 4, 16, 51, 154, 418, 1098, ...],
%e A089107 [0, 1, 5, 20, 70, 222, 654, 1817, ...],
%e A089107 [0, 1, 5, 24, 89, 309, 967, 2871, ...],
%e A089107 ...........
%p A089107 with(numtheory): m := proc(r,j) d := divisors(r): f := z->1/(1-z-z^2): W := (1/r)*z*sum(mobius(d[i])*f(z^d[i])^(r/d[i]),i=1..nops(d)): Wser := simplify(series(W,z=0,30)): coeff(Wser,z^j) end: seq(seq(m(n-q+1,q),q=1..n),n=1..17); # for the sequence read by antidiagonals
%p A089107 with(numtheory): f := z->1/(1-z-z^2): m := proc(r,j) d := divisors(r): W := (1/r)*z*sum(mobius(d[i])*f(z^d[i])^(r/d[i]),i=1..nops(d)): Wser := simplify(series(W,z=0,80)): coeff(Wser,z^j) end: matrix(10,10,m); # for the square array
%Y A089107 Adjacent sequences: A089104 A089105 A089106 this_sequence A089108 A089109 A089110
%Y A089107 Sequence in context: A025691 A099173 A108456 this_sequence A117398 A071486 A046695
%K A089107 nonn,tabl,easy
%O A089107 1,6
%A A089107 njas, Dec 05 2003
%E A089107 Edited by Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 06 2004

%I A117398
%S A117398 0,1,0,0,2,0,1,2,3,0,3,4,5,4,0,9,14,15,9,5,0,33,68,65,34,14,6,0,153,404,359,174,
%T A117398 63,20,7,0,873,2804,2375,1098,371,104,27,8,0,5913,22244,18215,8154,2639,692,159,35,9,
%U A117398 0,46233,198644,158615,69354,21791,5480,1179,230,44,10,0
%V A117398 0,1,0,0,2,0,-1,2,3,0,-3,4,5,4,0,-9,14,15,9,5,0,-33,68,65,34,14,6,0,-153,404,359,174,
%W A117398 63,20,7,0,-873,2804,2375,1098,371,104,27,8,0,-5913,22244,18215,8154,2639,692,159,35,9,
%X A117398 0,-46233,198644,158615,69354,21791,5480,1179,230,44,10,0
%N A117398 Matrix log of triangle A117396.
%C A117398 Column 0 contains negative of sequence A007489.
%e A117398 Triangle begins:
%e A117398 0;
%e A117398 1,0;
%e A117398 0,2,0;
%e A117398 -1,2,3,0;
%e A117398 -3,4,5,4,0;
%e A117398 -9,14,15,9,5,0;
%e A117398 -33,68,65,34,14,6,0;
%e A117398 -153,404,359,174,63,20,7,0;
%e A117398 -873,2804,2375,1098,371,104,27,8,0;
%e A117398 -5913,22244,18215,8154,2639,692,159,35,9,0; ...
%o A117398 (PARI) {T(n,k)=local(M=matrix(n+4,n+4,r,c,if(r>=c,if(r==c+1,-c,1))), L=sum(m=1,n+4,(M^0-M)^m/m));L[n+1,k+1]}
%Y A117398 Cf. A117396, A007489 (column 0), A117399 (column 1).
%Y A117398 Adjacent sequences: A117395 A117396 A117397 this_sequence A117399 A117400 A117401
%Y A117398 Sequence in context: A099173 A108456 A089107 this_sequence A071486 A046695 A071433
%K A117398 sign,tabl
%O A117398 0,5
%A A117398 Paul D. Hanna (pauldhanna(AT)juno.com), Mar 11 2006

%I A071486
%S A071486 1,2,0,1,2,3,1,2,3,1,4,3,0,4,3,1,4,2,1,3,2,1,0,2,1,4,2,6,4,1,6,4,2,
%T A071486 6,1,2,0,1,2,3
%N A071486 Sprague-Grundy values for octal game .224.
%D A071486 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982; see Chapter 4.
%Y A071486 Adjacent sequences: A071483 A071484 A071485 this_sequence A071487 A071488 A071489
%Y A071486 Sequence in context: A108456 A089107 A117398 this_sequence A046695 A071433 A110657
%K A071486 nonn
%O A071486 1,2
%A A071486 njas and Sue Pope (pope(AT)research.att.com), May 29 2002

%I A046695
%S A046695 0,0,1,2,0,1,2,3,1,2,3,4,0,3,4,2,1,3,2,1,0,2,1,4,5,1,4,5,1,2,0,1,2,3,1,
%T A046695 2,3,4,2,3,4,2,3,4,2,1,0,2,8,4,5,3,4,5,6,2,5,1,2,3,1,2,3,4,2,3,4,2,3,4,
%U A046695 2,3,0,2,3,4,5,3,4,5,6,4,5,6,2,3,1,2,3,4,2,3,4,2,3,4,2,3,0,2,3,4
%N A046695 Sprague-Grundy values for the game Couples-are-Forever (and for octal games .6, .601, .61, .611, .62, .621, .63, .631).
%D A046695 I. Caines et al., Periods in taking and splitting games, Amer. Math. Monthly, 106 (1999), 359-361.
%Y A046695 a(n) = A071433(n-2).
%Y A046695 Adjacent sequences: A046692 A046693 A046694 this_sequence A046696 A046697 A046698
%Y A046695 Sequence in context: A089107 A117398 A071486 this_sequence A071433 A110657 A071512
%K A046695 nonn
%O A046695 0,4
%A A046695 njas
%E A046695 Edited and extended by Christian G. Bower (bowerc(AT)usa.net), Oct 22 2002

%I A071433
%S A071433 1,2,0,1,2,3,1,2,3,4,0,3,4,2,1,3,2,1,0,2,1,4,5,1,4,5,1,2,0,1,2,3,1,2,3,
%T A071433 4,2,3,4,2,3,4,2,1,0,2,8,4,5,3,4,5,6,2,5,1,2,3,1,2,3,4,2,3,4,2,3,4,2,3,
%U A071433 0,2,3,4,5,3,4,5,6,4,5,6,2,3,1,2,3,4,2,3,4,2,3,4,2,3,0,2,3,4,5,3
%N A071433 Sprague-Grundy values for octal game .37.
%C A071433 Octal games .6, .601, .61, .611, .62, .621, .63, .631 (Officers) have values a(n-1).
%D A071433 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982; see Chapter 4, pp. 96, 102.
%D A071433 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Second Edition, Vol. 1, A K Peters, 2001, pp. 102, 109.
%D A071433 I. Caines et al., Periods in taking and splitting games, Amer. Math. Monthly, 106 (1999), 359-361.
%H A071433 Achim Flammenkamp, <a href="http://www.uni-bielefeld.de/~achim/octal_sparse.html">Octal games</a>
%Y A071433 a(n) = A046695(n+1).
%Y A071433 Adjacent sequences: A071430 A071431 A071432 this_sequence A071434 A071435 A071436
%Y A071433 Sequence in context: A117398 A071486 A046695 this_sequence A110657 A071512 A080018
%K A071433 nonn
%O A071433 1,2
%A A071433 njas and Sue Pope (pope(AT)research.att.com), May 29 2002
%E A071433 Edited and extended by Christian G. Bower (bowerc(AT)usa.net), Oct 22 2002

%I A110657
%S A110657 0,1,2,0,1,2,3,1,2,3,4,2,3,4,5,3,4,5,6,4,5,6,7,5,6,7,8,6,7,8,9,7,8,9,10,
%T A110657 8,9,10,11,9,10,11,12,10,11,12,13,11,12,13,14,12,13,14,15,13,14,15,16,
%U A110657 14,15,16,17,15,16,17,18,16,17,18,19,17,18,19,20,18,19,20,21,19,20,21
%N A110657 A028242(A028242(n)).
%C A110657 A110658(n) = A028242(a(n)) = a(A028242(n)).
%F A110657 a(n) = floor(n/4) + (n mod 4) mod 3.
%Y A110657 Cf. A110655.
%Y A110657 Adjacent sequences: A110654 A110655 A110656 this_sequence A110658 A110659 A110660
%Y A110657 Sequence in context: A071486 A046695 A071433 this_sequence A071512 A080018 A079686
%K A110657 nonn
%O A110657 0,3
%A A110657 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 05 2005

%I A071512
%S A071512 1,2,0,1,2,3,1,2,3,4,5,3,4,5,3,2,1,3,2,1,0,2,5,4,7,5,4,7,6,8,9,2,0,
%T A071512 1,2,3,9,6,7,4
%N A071512 Sprague-Grundy values for octal game .604.
%D A071512 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982; see Chapter 4.
%Y A071512 Adjacent sequences: A071509 A071510 A071511 this_sequence A071513 A071514 A071515
%Y A071512 Sequence in context: A046695 A071433 A110657 this_sequence A080018 A079686 A005813
%K A071512 nonn
%O A071512 1,2
%A A071512 njas and Sue Pope (pope(AT)research.att.com), May 29 2002

%I A080018
%S A080018 1,0,1,0,0,2,0,1,2,3,1,2,10,6,5,4,20,28,44,16,8,29,104,207,180,151,36,
%T A080018 13,206,775,1288,1407,830,437,76,21,1708,6140,10366,10384,7298,3100,
%U A080018 1138,152,34,15702,55427,91296,92896,63140,31278,10048,2744,294,55
%N A080018 Triangle of coefficients of polynomials P(n; x) = Permanent(M), where M=[m(i,j)] is n X n matrix defined by m(i,j)=x if -1<=i-j<=1 else m(i,j)=1.
%e A080018 1; 0,1; 0,0,2; 0,1,2,3; 1,2,10,6,5; 4,20,28,44,16,8;...
%e A080018 P(4; x) = Permanent(MATRIX([[x, x, 1, 1], [x, x, x, 1], [1, x, x, x], [1, 1, x, x]])) = 1+2*x+10*x^2+6*x^3+5*x^4.
%Y A080018 Row sums = A000142, first column = A001883, second column = A001884, third column = A001885, fourth column = A001886.
%Y A080018 Adjacent sequences: A080015 A080016 A080017 this_sequence A080019 A080020 A080021
%Y A080018 Sequence in context: A071433 A110657 A071512 this_sequence A079686 A005813 A049262
%K A080018 nonn,tabl
%O A080018 0,6
%A A080018 Vladeta Jovovic, Vladimir Baltic, (vladeta(AT)Eunet.yu), Jan 20 2003

%I A079686
%S A079686 0,1,0,1,1,2,0,1,2,3,1,3,0
%N A079686 Start with 0; at n-th step, write down what is in the sequence so far.
%Y A079686 Cf. A005150. For other versions see A051120, A079668, A079685.
%Y A079686 Adjacent sequences: A079683 A079684 A079685 this_sequence A079687 A079688 A079689
%Y A079686 Sequence in context: A110657 A071512 A080018 this_sequence A005813 A049262 A119464
%K A079686 nonn
%O A079686 0,6
%A A079686 njas (njas(AT)research.att.com), Jan 26 2003

%I A005813
%S A005813 1,0,0,0,0,0,1,0,0,0,1,0,2,0,1,2,3,1,4,2,5,5,7,4,10,8,12,
%T A005813 12,16,13,24,21,27,27,35,34,48,45,54,57,72,70,90,88,104,
%U A005813 112,132,132,159,162,188,199,228,230,270,281,316,333,373
%N A005813 Molien series for 6-dimensional complex representation of double cover of J2.
%D A005813 J. H. Conway and N. J. A. Sloane, circa 1977.
%H A005813 <a href="http://www.research.att.com/~njas/sequences/Sindx_Mo.html#Molien">Index entries for Molien series</a>
%p A005813 p/q = 1+x^12+x^20+2*x^24+x^28+..., where
%p A005813 p := x^140+x^110+x^108+x^106+2*x^104+2*x^102+3*x^100+3*x^98+3*x^96+3*x^94+4*x^92+
%p A005813 4*x^90+4*x^88+4*x^86+4*x^84+4*x^82+4*x^80+4*x^78+3*x^76+4*x^74+3*x^72+4*x^70+
%p A005813 3*x^68+4*x^66+3*x^64+4*x^62+4*x^60+4*x^58+4*x^56+4*x^54+4*x^52+4*x^50+4*x^48+
%p A005813 3*x^46+3*x^44+3*x^42+3*x^40+2*x^38+2*x^36+x^34+x^32+x^30+1;
%p A005813 q := (1-x^12)*(1-x^20)*(1-x^24)*(1-x^28)*(1-x^30)*(1-x^32);
%Y A005813 Adjacent sequences: A005810 A005811 A005812 this_sequence A005814 A005815 A005816
%Y A005813 Sequence in context: A071512 A080018 A079686 this_sequence A049262 A119464 A107017
%K A005813 nonn,easy,nice
%O A005813 0,13
%A A005813 njas

%I A049262
%S A049262 1,2,0,1,2,3,1,5,6,7,8,0,1,2,3,4,5,6,1,5,9,6,5,12,13,14,0,1,2,3,1,5,6,
%T A049262 6,8,9,1,5,3,13,6,15,1,17,18,19,20,0,1,2,3,4,2,6,7,8,6,10,8,12,1,2,9,
%U A049262 13,17,6,1,5,21,22,11,24,25,26,0,1,2,3,1,5,6,6,5,9,10,6,3,13,6,15,1,5
%N A049262 Smallest nonnegative value taken on by nx^2 - 3y^2 for an infinite number of integer pairs (x, y).
%Y A049262 Adjacent sequences: A049259 A049260 A049261 this_sequence A049263 A049264 A049265
%Y A049262 Sequence in context: A080018 A079686 A005813 this_sequence A119464 A107017 A049260
%K A049262 nonn
%O A049262 1,2
%A A049262 David W. Wilson (davidwwilson(AT)comcast.net)

%I A119464
%S A119464 0,0,0,0,1,0,0,1,0,2,0,1,2,3,3,0,0,1,0,2,0,1,2,3,3,0,4,0,1,4,5,5,0,2,4,
%T A119464 6,6,0,1,2,3,3,4,5,5,6,6,7,7,7,7,0,0,1,0,2,0,1,2,3,3,0,4,0,1,4,5,5,0,2,
%U A119464 4,6,6,0,1,2,3,3,4,5,5,6,6,7,7,7,7,0,8,0,1,8,9,9,0,2,8,10,10,0,1,2,3,3
%N A119464 Consider the standard game of Nim with 3 heaps and make a list of the losing positions (x,y,z) with x <= y <= z sorted by sum, ties broken by putting smallest value of x first, then y, then z; sequence gives x values.
%C A119464 Thanks to Ray Chandler (rayjchandler(AT)sbcglobal.net) for help in clarifying the relation between this sorted list and the one in A080593.
%e A119464 The triples with sum <= 20 (this sequence is the first column) are:
%e A119464 0 0 0
%e A119464 0 1 1
%e A119464 0 2 2
%e A119464 0 3 3
%e A119464 1 2 3
%e A119464 0 4 4
%e A119464 0 5 5
%e A119464 1 4 5
%e A119464 0 6 6
%e A119464 2 4 6
%e A119464 0 7 7
%e A119464 1 6 7
%e A119464 2 5 7
%e A119464 3 4 7
%e A119464 3 5 6
%e A119464 0 8 8
%e A119464 0 9 9
%e A119464 1 8 9
%e A119464 0 10 10
%Y A119464 Cf. A119465, A119466 give the y and z values; A080593, A080594, A080595 give the same values sorted in a different way.
%Y A119464 Adjacent sequences: A119461 A119462 A119463 this_sequence A119465 A119466 A119467
%Y A119464 Sequence in context: A079686 A005813 A049262 this_sequence A107017 A049260 A053186
%K A119464 easy,nonn
%O A119464 1,10
%A A119464 Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 21 2006

%I A107017
%S A107017 0,0,0,1,0,1,2,0,1,2,3,3,0,1,2,3,3,5,5,5,0,1,2,3,3,5,5,5,8,8,8,8,8,0,1,
%T A107017 2,3,3,5,5,5,8,8,8,8,8,13,13,13,13,13,13,13,13,0,1,2,3,3,5,5,5,8,8,8,8,
%U A107017 8,13,13,13,13,13,13,13,13,21,21,21,21,21,21,21,21,21,21,21,21,21,0,1,2
%N A107017 Second largest term in Zeckendorf representation of n, a(n)=0 if n itself is a Fibonacci number.
%C A107017 a(A000045(n)) = 0;
%C A107017 a(A000045(n) + k) = A087172(k) with 0 < k < A000045(n-1).
%H A107017 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ZeckendorfRepresentation.html">Zeckendorf Representation</a>
%H A107017 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>
%e A107017 n = 20 = 13+5+2: a(20) = 5.
%Y A107017 Cf. A007895.
%Y A107017 Adjacent sequences: A107014 A107015 A107016 this_sequence A107018 A107019 A107020
%Y A107017 Sequence in context: A005813 A049262 A119464 this_sequence A049260 A053186 A066628
%K A107017 nonn
%O A107017 1,7
%A A107017 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 09 2005

%I A049260
%S A049260 0,1,2,0,1,2,3,4,0,1,2,3,1,5,6,0,1,2,2,4,3,2,7,8,0,1,2,3,1,5,3,7,8,9,
%T A049260 10,0,1,2,3,4,1,6,2,7,5,5,11,12,0,1,2,3,1,2,6,7,3,1,2,11,1,13,14,0,1,2,
%U A049260 2,4,5,5,7,8,1,1,11,3,7,14,15,16,0,1,2,3,1,2,6,7,1,9,3,7,3,5,14,15,1
%N A049260 Smallest nonnegative value taken on by nx^2 - y^2 for an infinite number of integer pairs (x, y).
%Y A049260 Adjacent sequences: A049257 A049258 A049259 this_sequence A049261 A049262 A049263
%Y A049260 Sequence in context: A049262 A119464 A107017 this_sequence A053186 A066628 A135317
%K A049260 nonn
%O A049260 1,3
%A A049260 David W. Wilson (davidwwilson(AT)comcast.net)

%I A053186
%S A053186 0,0,1,2,0,1,2,3,4,0,1,2,3,4,5,6,0,1,2,3,4,5,6,7,8,0,1,2,3,4,5,6,7,8,9,
%T A053186 10,0,1,2,3,4,5,6,7,8,9,10,11,12,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,0,
%U A053186 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12
%N A053186 Square excess of n: difference between n and largest square <= n.
%D A053186 S. H. Weintraub, An interesting recursion, Amer. Math. Monthly, 111 (No. 6, 2004), 528-530.
%H A053186 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a000196.gif">Illustration of A000196, A048760, A053186</a>
%H A053186 M. Somos, <a href="http://www.research.att.com/~njas/sequences/a073189.txt">Sequences used for indexing triangular or square arrays</a>
%F A053186 a(n) = n-A048760(n) =n-floor(sqrt(n))^2
%p A053186 S:=proc(n) if issqr(n) then RETURN(0); fi; n-(floor(sqrt(n)))^2; end;
%o A053186 (PARI) a(n)=if(n<0,0,n-sqrtint(n)^2)
%Y A053186 Cf. A002262, A048760. A071797(n)=1+a(n-1).
%Y A053186 Adjacent sequences: A053183 A053184 A053185 this_sequence A053187 A053188 A053189
%Y A053186 Sequence in context: A119464 A107017 A049260 this_sequence A066628 A135317 A115218
%K A053186 easy,nonn
%O A053186 0,4
%A A053186 Henry Bottomley (se16(AT)btinternet.com), Mar 01 2000

%I A066628
%S A066628 0,0,0,0,1,0,1,2,0,1,2,3,4,0,1,2,3,4,5,6,7,0,1,2,3,4,5,6,7,8,9,10,11,
%T A066628 12,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,
%U A066628 7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30
%N A066628 a(n) = n - the largest Fibonacci number <= n.
%t A066628 f[n_] := Block[{k = 1}, While[Fibonacci[k] <= n, k++ ]; Return[n - Fibonacci[k - 1]]]; Table[ f[n], {n, 0, 100} ]
%Y A066628 Cf. A000045.
%Y A066628 Adjacent sequences: A066625 A066626 A066627 this_sequence A066629 A066630 A066631
%Y A066628 Sequence in context: A107017 A049260 A053186 this_sequence A135317 A115218 A023858
%K A066628 nonn
%O A066628 0,8
%A A066628 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Dec 25 2001
%E A066628 Corrected and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 28 2001

%I A135317
%S A135317 0,1,2,0,1,2,3,4,0,3,4,5,1,2,3,6,0,1,4,5,6,2,5,6,7,8,0,7,8,9,3,4,5,1,2,
%T A135317 3,6,7,10,4,7,8,0,1,2,9,10,11,5,8,9,12,6,7,10,11,12,8,9,10,0,3,4,13,1,2,
%U A135317 5,6,11,3,4,9,14,0,1,10,11,12,2,7,8,13,5,6,9,12,13,7,10,11,14,15,16,14
%N A135317 Sequence yielding an ordering of N*N derived from a family of recurrences. For any integer k define h(k,1)=1 and for n>1 define h(k,n)=h(k,n-1)+2*((-h(k,n-1)mod n)where "r mod s" denotes least nonnegative residue of r modulo s [informally, h(k,n) is got by "reflecting" h(k,n-1)in the least multiple of n that is >=h(k,n-1)]. Then for fixed k>=0 there are integers a(k), b(k), n(k) such that for all n>n(k) we have h(k,2*n+1)-h(k,2*n)=2*a(k)and h(k,2*n+2)-h(k,2*n+1)=2*b(k). For all k we have a(2*k+1)=a(2*k) and b(2*k+1)=1+b(2*k). Moreover b(2*k) is even for all k. The function k->(a(2*k),b(2*k)/2) is a bijection from the nonnegative integers N to N*N. It is "monotone" in the sense that k<=k' whenever a(2*k)<=a(2*k') and b(2*k)<=b(2*k'). The sequence given above is a(2*k).
%C A135317 The results on which the definition is based are not yet proved, but they are plausible and overwhelmingly supported by numerical evidence. I am working on a proof.
%C A135317 For each fixed n, k->h(k,n) is a bijection Z->Z (this is easy!). However for k<0 the sequence h(k,n) does not have the pseudo-periodic property we have used in defining a(k) and b(k).
%C A135317 n(k) appears to be O(sqrt k).
%e A135317 h(18,n) for n>=1 goes 18,18,18,22,28,32,38,42,48...so we can take n(18)=2, then 2*a(18)=28-22=38-32=48-42=...=6, so a(18)=3 and similarly b(18)=2.
%Y A135317 Adjacent sequences: A135314 A135315 A135316 this_sequence A135318 A135319 A135320
%Y A135317 Sequence in context: A049260 A053186 A066628 this_sequence A115218 A023858 A011118
%K A135317 nonn,uned
%O A135317 0,3
%A A135317 Alex Abercrombie (alexabercrombie(AT)hotmail.co.uk), Feb 15 2008

%I A115218
%S A115218 0,0,1,2,0,1,2,3,4,5,6,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,0,