The Database of Integer Sequences, Part 12
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 A129177
%S A129177 1,1,1,2,2,1,1,6,6,3,5,2,1,1,24,24,12,20,14,10,7,5,2,1,1,120,120,60,100,
%T A129177 70,74,59,37,30,19,15,7,5,2,1,1,720,720,360,600,420,444,474,342,240,214,
%U A129177 160,116,89,49,36,25,15,7,5,2,1,1,5040,5040,2520,4200,2940,3108,3318
%N A129177 Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} such that w(p)=k (n>=1; 0<=k<=n(n-1)/2); w(p) is defined (by Edelman, Simion and White) in the following way: if p = (c[1])(c[2])... is expressed in standard cycle form ( i.e. cycles ordered by increasing smallest elements with each cycle written with its smallest element in the first position), then w(p)=0*|c[1]| + 1*|c[2]| + 2*|c[3]| +..., where |c[j]| denotes the number of entries in the cycle c[j].
%C A129177 Row n has 1+n(n-1)/2 terms. Row sums are the factorials (A000142). T(n,0)=T(n,1)=(n-1)! for n>=2. T(n,2)=(n-1)!/2=A001710(n-1) for n>=3. Sum(k*T(n,k),k>=0)=A067318(n).
%D A129177 P. H. Edelman, R. Simion and D. White, Partition statistics on permutations. Discrete Math. 99 (1992), 63-68.
%D A129177 M. Shattuck, Parity theorems for statistics on permutations and Catalan words, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 5, Paper A07, 2005.
%F A129177 Generating polynomial of row n is P[n](t)=Product(i + t^i, i=0..n-1).
%e A129177 T(4,2)=3 because we have w(1423)=w((1)(243))=0*1+1*3=3, w(1342)=w((1)(234))=0*1+1*3=3 and w(2134)=w((12)(3)(4))=0*2+1*1+2*1=3.
%e A129177 Triangle starts:
%e A129177 1;
%e A129177 1,1;
%e A129177 2,2,1,1;
%e A129177 6,6,3,5,2,1,1;
%e A129177 24,24,12,20,14,10,7,5,2,1,1;
%p A129177 for n from 1 to 8 do P[n]:=sort(expand(product(i+t^i,i=0..n-1))) od: for n from 1 to 8 do seq(coeff(P[n],t,j),j=0..n*(n-1)/2) od; # yields sequence in triangular form
%Y A129177 Cf. A000142, A001710, A067318.
%Y A129177 Adjacent sequences: A129174 A129175 A129176 this_sequence A129178 A129179 A129180
%Y A129177 Sequence in context: A086610 A114626 A124773 this_sequence A127452 A135879 A139331
%K A129177 nonn,tabf
%O A129177 1,4
%A A129177 Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 11 2007
%I A127452
%S A127452 1,1,1,2,2,1,1,6,6,4,4,2,1,1,24,24,18,18,12,8,8,4,2,1,1,120,120,96,96,
%T A127452 72,54,54,36,24,16,16,8,4,2,1,1,720,720,600,600,480,384,384,288,216,162,
%U A127452 162,108,72,48,32,32,16,8,4,2,1,1
%N A127452 Triangle, read by rows of n*(n+1)/2 + 1 terms, generated by the recurrence: start with a single '1' in row 0; row n+1 is generated from row n by first inserting zeros at positions {(m+1)*(m+2)/2 - 1, m>=0} in row n and then taking the partial sums in reverse order.
%C A127452 The first column equals the factorials. Triangle A127420 is generated by a similar recurrence.
%F A127452 Sum_{k=0..n*(n+1)/2} k*T(n,k) = A018927(n+1) = Sum_{k=0..n} k*k!*{(k+1)^(n-k+1)-k^(n-k+1)}.
%F A127452 T(n,k) = (n-t)! * (n-t)^(k - t*(t+1)/2) * (n-t+1)^(t-k + t*(t+1)/2) where t=floor((sqrt(8*k+1)-1)/2). Also, Sum_{j=k*(k+1)/2..(k+1)*(k+2)/2-1} T(n,j) = A047969(n-k,k) = (n-k)!*((n-k+1)^(k+1)-(n-k)^(k+1)).
%e A127452 The triangle begins:
%e A127452 1;
%e A127452 1, 1;
%e A127452 2, 2, 1, 1;
%e A127452 6, 6, 4, 4, 2, 1, 1;
%e A127452 24, 24, 18, 18, 12, 8, 8, 4, 2, 1, 1;
%e A127452 120, 120, 96, 96, 72, 54, 54, 36, 24, 16, 16, 8, 4, 2, 1, 1;
%e A127452 720, 720, 600, 600, 480, 384, 384, 288, 216, 162, 162, 108, 72, 48, 32, 32, 16, 8, 4, 2, 1, 1;
%e A127452 ...
%e A127452 The recurrence is illustrated by the following examples.
%e A127452 Start with a single '1' in row 0.
%e A127452 To get row 1, insert 0 in row 0 at position 0,
%e A127452 and take partial sums in reverse order:
%e A127452 0,_1;
%e A127452 1,_1;
%e A127452 To get row 2, insert 0 in row 1 at positions [0,2],
%e A127452 and take partial sums in reverse order:
%e A127452 0,_1,_0,_1;
%e A127452 2,_2,_1,_1;
%e A127452 To get row 3, insert 0 in row 2 at positions [0,2,5],
%e A127452 and take partial sums in reverse order:
%e A127452 0,_2,_0,_2,_1,_0,_1;
%e A127452 6,_6,_4,_4,_2,_1,_1;
%e A127452 To get row 4, insert 0 in row 3 at positions [0,2,5,9],
%e A127452 and take partial sums in reverse order:
%e A127452 _0,__6,__0,__6,__4,_0,_4,_2,_1,_0,_1;
%e A127452 24,_24,_18,_18,_12,_8,_8,_4,_2,_1,_1;
%e A127452 etc.
%e A127452 Continuing in this way generates the factorials in the first column.
%o A127452 (PARI) {T(n,k)=if(n<0|k<0,0,if(n==0&k==0,1,if(k==0,n!,if(issquare(8*k+1),T(n,k-1),T(n,k-1)-T(n-1,k-(sqrtint(8*k+1)+1)\2)))))}
%o A127452 (PARI) {T(n,k)=local(t=(sqrtint(8*k+1)-1)\2);(n-t)!*(n-t)^(k-t*(t+1)/2)*(n-t+1)^(t-k+t*(t+1)/2)}
%Y A127452 Cf. A018927, A127420.
%Y A127452 Cf. A047969.
%Y A127452 Adjacent sequences: A127449 A127450 A127451 this_sequence A127453 A127454 A127455
%Y A127452 Sequence in context: A114626 A124773 A129177 this_sequence A135879 A139331 A090441
%K A127452 nonn,tabl
%O A127452 0,4
%A A127452 Paul D. Hanna (pauldhanna(AT)juno.com), Jan 15 2007
%I A135879
%S A135879 1,1,1,2,2,1,1,6,6,4,4,2,2,1,25,25,19,19,13,13,9,5,5,3,1,1,138,138,113,
%T A135879 113,88,88,69,50,50,37,24,24,15,10,5,5,2,1,970,970,832,832,694,694,581,
%U A135879 468,468,380,292,292,223,173,123,123,86,62,38,38,23,13,8,3,3,1,8390
%N A135879 Triangle, read by rows of A135901(n) terms, where row n+1 is generated from row n by inserting zeros at positions [(m+3)^2/4 - 2], as m=0,1,2,3,..., and then taking partial sums from right to left, starting with a single 1 in row 0.
%C A135879 Column 0 is A135881 which equals column 0 of square array A135878 and also equals column 0 of triangle A135880. Compare to square array A135878, which is generated by a complementary process. An interesting variant is triangle A135877 in which column 0 equals the double factorials (A001147).
%e A135879 Triangle begins:
%e A135879 1;
%e A135879 1, 1;
%e A135879 2, 2, 1, 1;
%e A135879 6, 6, 4, 4, 2, 2, 1;
%e A135879 25, 25, 19, 19, 13, 13, 9, 5, 5, 3, 1, 1;
%e A135879 138, 138, 113, 113, 88, 88, 69, 50, 50, 37, 24, 24, 15, 10, 5, 5, 2, 1;
%e A135879 970, 970, 832, 832, 694, 694, 581, 468, 468, 380, 292, 292, 223, 173, 123, 123, 86, 62, 38, 38, 23, 13, 8, 3, 3, 1;
%e A135879 8390, 8390, 7420, 7420, 6450, 6450, 5618, 4786, 4786, 4092, 3398, 3398, 2817, 2349, 1881, 1881, 1501, 1209, 917, 917, 694, 521, 398, 275, 275, 189, 127, 89, 51, 51, 28, 15, 7, 4, 1, 1;
%e A135879 There are A135901(n) number of terms in row n.
%e A135879 To generate the triangle, start with a single 1 in row 0,
%e A135879 and then obtain row n+1 from row n by inserting zeros at
%e A135879 positions {[(m+3)^2/4 - 2], m=0,1,2,...}, and then
%e A135879 taking reverse partial sums (i.e., summing from right to left).
%e A135879 Start with row 0, insert a zero in front of the '1' at position 0:
%e A135879 [0,1];
%e A135879 take reverse partial sums to get row 1:
%e A135879 [1,1];
%e A135879 insert zeros at positions [0,2]:
%e A135879 [0,1,0,1];
%e A135879 take reverse partial sums to get row 2:
%e A135879 [2,2,1,1];
%e A135879 insert zeros at positions [0,2,4]:
%e A135879 [0,2,0,2,0,1,1];
%e A135879 take reverse partial sums to get row 3:
%e A135879 [6,6,4,4,2,2,1];
%e A135879 insert zeros at positions [0,2,4,7]:
%e A135879 [0,6,0,6,0,4,4,0,2,2,0,1];
%e A135879 take reverse partial sums to get row 4:
%e A135879 [25,25,19,19,13,13,9,5,5,3,1,1];
%e A135879 insert zeros at positions [0,2,4,7,10,14]:
%e A135879 [0,25,0,25,0,19,19,0,13,13,0,9,5,5,0,3,1,1];
%e A135879 take reverse partial sums to get row 5:
%e A135879 [138,138,113,113,88,88,69,50,50,37,24,24,15,10,5,5,2,1].
%e A135879 Triangle A135880 begins:
%e A135879 1;
%e A135879 1, 1;
%e A135879 2, 2, 1;
%e A135879 6, 7, 3, 1;
%e A135879 25, 34, 15, 4, 1;
%e A135879 138, 215, 99, 26, 5, 1;
%e A135879 970, 1698, 814, 216, 40, 6, 1; ...
%e A135879 and is generated by matrix powers of itself.
%o A135879 (PARI) {T(n,k)=local(A=[1],B);if(n>0,for(i=1,n,m=1;B=[]; for(j=1,#A,if(j+m-1==floor((m+2)^2/4)-1,m+=1;B=concat(B,0));B=concat(B,A[j])); A=Vec(Polrev(Vec(Pol(B)/(1-x+O(x^#B)))))));if(k+1>#A,0,A[k+1])}
%Y A135879 Cf. A135881, A135901, A135878, A135880; variants: A135877, A127452, A125781.
%Y A135879 Adjacent sequences: A135876 A135877 A135878 this_sequence A135880 A135881 A135882
%Y A135879 Sequence in context: A124773 A129177 A127452 this_sequence A139331 A090441 A107876
%K A135879 nonn,tabl
%O A135879 0,4
%A A135879 Paul D. Hanna (pauldhanna(AT)juno.com), Dec 14 2007
%I A139331
%S A139331 1,1,1,1,1,1,1,2,2,1,1,6,12,6,1,1,12,72,72,12,1,1,20,240,720,240,20,1,1,
%T A139331 30,600,3600,3600,600,30,1,1,56,1680,16800,33600,16800,1680,56,1,1,72,
%U A139331 4032,60480,201600,201600,60480,4032,72,1
%N A139331 Triangle read by rows: ConvOffsStoT transform of A103505.
%e A139331 First few rows of the triangle are:
%e A139331 1;
%e A139331 1, 1;
%e A139331 1, 1, 1;
%e A139331 1, 2, 2, 1;
%e A139331 1, 6, 12, 6, 1;
%e A139331 1, 12, 72, 72, 12, 1;
%e A139331 1, 20, 240, 720, 240, 20, 1;
%e A139331 1, 30, 600, 3600, 3600, 600, 30, 1;
%e A139331 ..
%e A139331 Row 4 = (1, 6, 12, 6, 1) since ConvOffs transform of (1, 1, 2, 6) = (1, 6, 12, 6, 1); where A103505 = (1, 1, 2, 6, 12, 20, 30, 56, 72,...).
%Y A139331 Cf. A103505.
%Y A139331 Adjacent sequences: A139328 A139329 A139330 this_sequence A139332 A139333 A139334
%Y A139331 Sequence in context: A129177 A127452 A135879 this_sequence A090441 A107876 A121554
%K A139331 nonn,tabl
%O A139331 0,8
%A A139331 Gary W. Adamson and Roger L. Bagula (qntmpkt(AT)yahoo.com), Apr 13 2008
%I A090441
%S A090441 1,1,1,1,1,1,1,2,2,1,1,6,12,6,1,1,24,144,144,24,1,1,120,2880,8640,2880,
%T A090441 120,1,1,720,86400,1036800,1036800,86400,720,1,1,5040,3628800,217728000,
%U A090441 870912000,217728000,3628800,5040,1,1,40320,203212800,73156608000
%N A090441 Symmetric triangle of certain normalized products of decreasing factorials.
%C A090441 Similar to, but different from, superfactorial Pascal triangle A009963.
%C A090441 A009963(n,m)= product((n-p)!,p=0..m-1)/superfac(m) with n>=m>=0 else 0.
%H A090441 W. Lang, First 9 rows.
%F A090441 a(n, m)=0 if n< m-1; a(n, m)=1 if m=0 or n=-1; a(n, m)= product((n-p)!, p=0..m-1)/superfac(m-1) if n>=0, 1<=m<=n+1, where superfac(n) := A000178(n), n>=0, (superfactorials).
%F A090441 Equals ConvOffsStoT transform of the factorials, A000142: (1, 1, 2, 6, 24,...); e.g., ConvOffs transform of (1, 1, 2, 6) = (1, 6, 12, 6, 1). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 21 2008
%e A090441 [1];[1,1];[1,1,1];[1,2,2,1];[1,6,12,6,1];..., rows for
%e A090441 n=-1,0,1,2,3,...
%Y A090441 Column sequences give: A000012 (powers of 1), A000142 (factorials), A010790, A090443-4, etc.
%Y A090441 Cf. A090445 (row sums), A090446 (alternating row sums).
%Y A090441 Adjacent sequences: A090438 A090439 A090440 this_sequence A090442 A090443 A090444
%Y A090441 Sequence in context: A127452 A135879 A139331 this_sequence A107876 A121554 A011296
%K A090441 nonn,easy,tabl,new
%O A090441 -1,8
%A A090441 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Dec 23 2003
%I A107876
%S A107876 1,1,1,1,1,1,2,2,1,1,7,7,3,1,1,37,37,15,4,1,1,268,268,106,26,5,1,1,2496,
%T A107876 2496,975,230,40,6,1,1,28612,28612,11100,2565,425,57,7,1,1,391189,
%U A107876 391189,151148,34516,5570,707,77,8,1,1,6230646,6230646,2401365,544423
%N A107876 Triangular matrix T, read by rows, that satisfies: [T^k](n,k) = T(n,k-1) for n>=k>0, or, equivalently, (column k of T^k) = SHIFT_LEFT(column k-1 of T) when zeros above the diagonal are ignored.
%C A107876 Remarkably, T equals the product of these triangular matrices: T = A107862^-1*A107867 = A107867^-1*A107870 = A107870^-1*A107873. T also satisfies: [T^-1](n,k) = -[T^k](n,k+1) for n>k>=0.
%C A107876 Column m of T^k is the number of subpartitions of the initial terms of the sequence (k-1)+n(m-1)+n(n-1)/2 (ignoring 0's above the diagonal). E.g. column 4 of T^3 is 1,3,15,106,975,.... The sequence above is 2,5,9,14,20,.... subp([]) = 1, subp([2]) = 3, subp([2,5]) = 15, subp([2,5,9]) = 106, etc. The matrix product of T^(k-1) * T computes the number of such subpartitions by looking at the first part index where the subpartition is maxed - for [2,5,9,14,20] the third term (9 maxed) has subp([1,4]) for the first two values (not maxed), times subp([5,11]) for the last two values (possibly maxed). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 26 2006
%F A107876 G.f. for column k: 1 = Sum_{j>=0} T(k+j, k)*x^j*(1-x)^(1+(k+j)*(k+j-1)/2-k*(k-1)/2). G.f. for column k of T^m: 1 = Sum_{j>=0} [T^m](k+j, k)*x^j*(1-x)^(m+(k+j)*(k+j-1)/2-k*(k-1)/2) where [T^m] is the m-th matrix power of T, for all m and k>=0. G.f. for column k of T^m: 1 = Sum_{j>=0} [T^m](k+j, k)*x^j/C(x)^(m-j+(k+j)*(k+j-1)/2-k*(k-1)/2) where C(x)=2/(1+sqrt(1-4*x)) is g.f. for A000108 (Catalan numbers).
%e A107876 G.f. for column 1:
%e A107876 1 = T(1,1)*(1-x)^1 + T(2,1)*x*(1-x)^2 + T(3,1)*x^2*(1-x)^4 + T(4,1)*x^3*(1-x)^7 + T(5,1)*x^4*(1-x)^11 + T(6,1)*x^5*(1-x)^16 +...
%e A107876 = 1*(1-x)^1 + 1*x*(1-x)^2 + 2*x^2*(1-x)^4 + 7*x^3*(1-x)^7 + 37*x^4*(1-x)^11 + 268*x^5*(1-x)^16 +...
%e A107876 G.f. for column 2:
%e A107876 1 = T(2,2)*(1-x)^1 + T(3,2)*x*(1-x)^3 + T(4,2)*x^2*(1-x)^6 + T(5,2)*x^3*(1-x)^10 + T(6,2)*x^4*(1-x)^15 + T(7,2)*x^5*(1-x)^21 +...
%e A107876 = 1*(1-x)^1 + 1*x*(1-x)^3 + 3*x^2*(1-x)^6 + 15*x^3*(1-x)^10 + 106*x^4*(1-x)^15 + 975*x^5*(1-x)^21 +...
%e A107876 Triangle T begins:
%e A107876 1;
%e A107876 1,1;
%e A107876 1,1,1;
%e A107876 2,2,1,1;
%e A107876 7,7,3,1,1;
%e A107876 37,37,15,4,1,1;
%e A107876 268,268,106,26,5,1,1;
%e A107876 2496,2496,975,230,40,6,1,1;
%e A107876 28612,28612,11100,2565,425,57,7,1,1;
%e A107876 391189,391189,151148,34516,5570,707,77,8,1,1; ...
%e A107876 where column 1 of T = SHIFT_LEFT(column 0 of T).
%e A107876 Matrix square T^2 begins:
%e A107876 1;
%e A107876 2,1;
%e A107876 3,2,1;
%e A107876 7,5,2,1;
%e A107876 26,19,7,2,1;
%e A107876 141,104,37,9,2,1;
%e A107876 1034,766,268,61,11,2,1; ...
%e A107876 Compare column 2 of T^2 with column 1 of T.
%e A107876 Matrix inverse begins:
%e A107876 1;
%e A107876 -1,1;
%e A107876 0,-1,1;
%e A107876 0,-1,-1,1;
%e A107876 0,-3,-2,-1,1;
%e A107876 0,-15,-9,-3,-1,1;
%e A107876 0,-106,-61,-18,-4,-1,1; ...
%e A107876 Compare column 1 of T^-1 with column 2 of T, and
%e A107876 compare column 2 of T^-1 with column 3 of T^2.
%o A107876 (PARI) {T(n,k)=polcoeff(1-sum(j=0,n-k-1, T(j+k,k)*x^j*(1-x+x*O(x^n))^(1+(k+j)*(k+j-1)/2-k*(k-1)/2)),n-k)} (PARI) {T(n,k,p)=polcoeff(1- sum(j=0,n-k-1,T(j+k,k,p)*x^j*(1-x+x*O(x^n))^(j*(j-1)/2+j*k+p)),n-k)}
%Y A107876 Cf. A107862, A107865, A107867, A107870, A107877 (column 1), A107878 (column 2), A107879 (column 3), A107880 (matrix square), A107884 (matrix cube), A107889 (matrix inverse).
%Y A107876 Cf. A115728, A115729.
%Y A107876 Adjacent sequences: A107873 A107874 A107875 this_sequence A107877 A107878 A107879
%Y A107876 Sequence in context: A135879 A139331 A090441 this_sequence A121554 A011296 A016739
%K A107876 nonn,tabl,nice
%O A107876 0,7
%A A107876 Paul D. Hanna (pauldhanna(AT)juno.com), Jun 04 2005
%I A121554
%S A121554 1,0,1,1,0,1,2,2,1,1,7,7,6,3,1,30,35,30,18,6,1,157,205,184,117,46,10,1,
%T A121554 972,1392,1304,874,381,101,15,1,6961,10764,10499,7355,3470,1052,197,21,
%U A121554 1,56660,93493,94668,68909,34622,11606,2542,351,28,1,516901,901900
%N A121554 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 1-cell columns (0<=k<=n). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.
%C A121554 Row sums are the factorials (A000142). T(n,0)=A001053(n). Sum(k*T(n,k), k=0..n)=A121555(n).
%D A121554 E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
%F A121554 The row generating polynomials are P(n,t)=Q(n,t,1), where Q(0,t,x)=1 and Q(n,t,x)=Q(n-1,t,1/t)+(tx+n-2)Q(n-1,t,1) for n>=1.
%e A121554 T(2,0)=1, T(2,1)=0, T(2,2)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having, respectively, 0 and 2 columns with exactly 1 cell.
%e A121554 Triangle starts:
%e A121554 1;
%e A121554 0,1;
%e A121554 1,0,1;
%e A121554 2,2,1,1;
%e A121554 7,7,6,3,1;
%e A121554 30,35,30,18,6,1;
%p A121554 Q[0]:=1: for n from 1 to 10 do Q[n]:=sort(expand(subs(x=1/t,Q[n-1])+(t*x+n-2)*subs(x=1,Q[n-1]))) od: for n from 0 to 10 do P[n]:=subs(x=1,Q[n]) od: for n from 0 to 10 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form
%Y A121554 Cf. A000142, A001053, A121555.
%Y A121554 Adjacent sequences: A121551 A121552 A121553 this_sequence A121555 A121556 A121557
%Y A121554 Sequence in context: A139331 A090441 A107876 this_sequence A011296 A016739 A039965
%K A121554 nonn,tabl
%O A121554 0,7
%A A121554 Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 08 2006
%I A011296
%S A011296 1,2,2,1,1,8,8,5,5,0,3,1,1,9,9,3,7,6,3,8,2,3,0,2,0,8,4,3,1,3,6,9,4,
%T A011296 6,5,5,7,1,5,1,1,6,8,6,1,1,7,9,9,2,7,2,1,5,0,5,7,1,1,0,0,0,8,8,8,2,
%U A011296 4,8,6,5,5,0,2,2,7,8,3,9,3,2,0,7,6,6,6,0,4,3,1,0,7,5,7,1,7,8,9,3,8
%N A011296 Decimal expansion of 12th root of 11.
%Y A011296 Adjacent sequences: A011293 A011294 A011295 this_sequence A011297 A011298 A011299
%Y A011296 Sequence in context: A090441 A107876 A121554 this_sequence A016739 A039965 A074942
%K A011296 nonn,cons
%O A011296 1,2
%A A011296 njas
%I A016739
%S A016739 2,2,1,1,18,2,2,7,1,6,1,1,4,1,32,4,1,1,1,3,1,1,1,2,1,4,
%T A016739 1,1,3,1,43,4,1,1,1,25,3,2,17,4,2,1,2,1,3,1,1,89,5,1,1,
%U A016739 1,3,1,1,2,15,3,1,49,1,1,2,35,23,1,7,2,2,3,1,2,2,1,74,1
%N A016739 Continued fraction for ln(11).
%H A016739 G. Xiao, Contfrac
%Y A016739 Adjacent sequences: A016736 A016737 A016738 this_sequence A016740 A016741 A016742
%Y A016739 Sequence in context: A107876 A121554 A011296 this_sequence A039965 A074942 A043754
%K A016739 nonn,cofr
%O A016739 1,1
%A A016739 njas
%I A039965
%S A039965 1,2,2,1,2,0,0,0,2,1,2,2,1,2,0,0,0,0,0,0,0,0,0,0,0,0,2,1,2,2,1,2,0,0,0,
%T A039965 2,1,2,2,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A039965 0,0,0,0,0,0,0,0,0,0,2,1,2,2,1,2,0,0,0,2,1,2,2,1,2,0,0,0,0,0,0,0,0,0,0
%N A039965 An example of a d-perfect sequence.
%H A039965 D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions
%H A039965 D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions
%F A039965 a(n) = ((-1)^(n+1)*A000108(n-1)) mod 3 - Christian G. Bower (bowerc(AT)usa.net), Jun 12 2005 - Christian G. Bower (bowerc(AT)usa.net), Jun 12 2005
%Y A039965 Adjacent sequences: A039962 A039963 A039964 this_sequence A039966 A039967 A039968
%Y A039965 Sequence in context: A121554 A011296 A016739 this_sequence A074942 A043754 A094022
%K A039965 nonn
%O A039965 1,2
%A A039965 njas
%E A039965 More terms from Christian G. Bower (bowerc(AT)usa.net), Jun 12 2005
%I A074942
%S A074942 1,1,2,2,1,2,0,1,0,1,1,1,0,0,2,2,1,0,0,2,0,1,1,2,2,0,0,0,1,2,0,1,2,1,0,
%T A074942 0,0,0,0,1,1,0,0,2,0,1,1,1,0,2,2,0,1,0,1,0,0,1,1,1,0,0,0,2,0,2,0,2,2,0,
%U A074942 1,0,0,0,1,0,0,0,0,2,0,1,1,0,1,0,2,1,1,0,0,2,0,1,0,2,0,0,0,1,1,2,0,0,0
%N A074942 phi(n) (mod 3).
%o A074942 (PARI) a(n)=eulerphi(n)%3
%Y A074942 Adjacent sequences: A074939 A074940 A074941 this_sequence A074943 A074944 A074945
%Y A074942 Sequence in context: A011296 A016739 A039965 this_sequence A043754 A094022 A128580
%K A074942 easy,nonn
%O A074942 1,3
%A A074942 Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 04 2002
%I A043754
%S A043754 0,2,2,1,2,0,1,0,3,0,4,1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0,0,1,0,1,1,
%T A043754 0,0,1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0,0,1,0,0,0,1,0,2,0,0,1,0,0,0
%N A043754 Let L(n) = n-th lucky number. Then a(1)=0; a(n) is largest number m such that all divisors of L(n)^m are lucky numbers.
%e A043754 To get a(3) we compute L(3)=7, Divisors of 7^2 are 1,7,49. (1,7,49)->lucky number, so a(3)=2.
%Y A043754 Adjacent sequences: A043751 A043752 A043753 this_sequence A043755 A043756 A043757
%Y A043754 Sequence in context: A016739 A039965 A074942 this_sequence A094022 A128580 A129402
%K A043754 more,nonn
%O A043754 0,2
%A A043754 Naohiro Nomoto (6284968128(AT)geocities.co.jp), Oct 08 2000
%E A043754 More terms from Sascha Kurz (sascha.kurz(AT)uni-bayreuth.de), Oct 14 2001
%I A094022
%S A094022 1,0,1,1,1,0,2,2,1,2,0,1,2,2,3,7,2,6,8,5,2,12,10,6,13,4,7,14,10,14,32,
%T A094022 12,24,36,22,13,50,36,26,56,22,30,62,40,51,114,46,79,129,76,54,170,114,
%U A094022 90,192,82,104,216,132,159,350,152,230,397,226,180,506,322,270,574,260
%V A094022 1,0,-1,1,-1,0,2,-2,-1,2,0,-1,2,-2,-3,7,-2,-6,8,-5,-2,12,-10,-6,13,-4,-7,14,-10,-14,32,
%W A094022 -12,-24,36,-22,-13,50,-36,-26,56,-22,-30,62,-40,-51,114,-46,-79,129,-76,-54,170,-114,
%X A094022 -90,192,-82,-104,216,-132,-159,350,-152,-230,397,-226,-180,506,-322,-270,574,-260
%N A094022 Expansion of et(q^2)eta(q^30)/(eta(q^3)eta(q^5)) in powers of q.
%F A094022 G.f. A(x) satisfies 0=f(A(x),A(x^2)) where f(u,v)=u^2-v+2v^2-2uv^2.
%F A094022 G.f. A(x) satisfies A(x)+A(-x)=2A(x^2)^2, (1-A(x))(1-A(-x))=1-A(x^2).
%F A094022 Euler transform of period 30 sequence [ -1,1,-1,1,0,0,-1,1,0,0,0,0,-1,2,-1,0,0,0,0,1,-1,0,0,1,-1,1,-1,0,0,0,...].
%o A094022 (PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^30+A)/eta(x^3+A)/eta(x^5+A), n))}
%Y A094022 Adjacent sequences: A094019 A094020 A094021 this_sequence A094023 A094024 A094025
%Y A094022 Sequence in context: A039965 A074942 A043754 this_sequence A128580 A129402 A134177
%K A094022 sign
%O A094022 1,7
%A A094022 Michael Somos, Apr 22 2004
%I A128580
%S A128580 1,1,2,2,1,2,0,2,0,0,2,0,3,1,2,2,2,4,0,0,0,0,2,0,3,0,2,4,0,2,0,2,0,0,0,
%T A128580 0,2,3,4,2,1,2,0,2,0,0,2,0,2,2,2,2,4,2,0,0,0,0,0,0,3,0,4,2,0,2,0,2,0,0,
%U A128580 0,0,4,3,2,2,0,4,0,2,0,0,4,0,1,0,2,6,2,2,0,0,0,0,2,0,2,0,2,2,0,4,0,0,0
%V A128580 1,-1,-2,2,1,-2,0,2,0,0,-2,0,3,-1,-2,2,2,-4,0,0,0,0,-2,0,3,0,-2,4,0,-2,0,2,0,0,0,0,2,
%W A128580 -3,-4,2,1,-2,0,2,0,0,-2,0,2,-2,-2,2,4,-2,0,0,0,0,0,0,3,0,-4,2,0,-2,0,2,0,0,0,0,4,-3,
%X A128580 -2,2,0,-4,0,2,0,0,-4,0,1,0,-2,6,2,-2,0,0,0,0,-2,0,2,0,-2,2,0,-4,0,0,0
%N A128580 Expansion of phi(q^3)* psi(q^4) -q* phi(q)* psi(q^12) in powers of q where phi(), psi() are Ramanujan theta functions.
%F A128580 a(n)=b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = (-1)^e, b(p^e) = e+1 if p == 1, 7 (mod 24), b(p^e) = (e+1)* (-1)^e if p == 5, 11 (mod 24), b(p^e) = (1+(-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
%F A128580 Euler transform of period 24 sequence [ -1, -2, 0, 0, -1, -1, -1, -1, 0, -2, -1, -2, -1, -2, 0, -1, -1, -1, -1, 0, 0, -2, -1, -2, ...].
%F A128580 a(12n+6)= a(12n+8)= a(12n+9)= a(12n+11)= 0.
%F A128580 G.f.: Product_{k>0} (1-x^(8k))* (1-x^(12k))^2/ ((1+x^k)* (1+x^(2k))^2* (1-x^(3k))* (1+x^(12k))).
%F A128580 G.f.: Sum_{k>=0} a(k)*x^(2*k+1) = Sum_{k>0} (x^k - x^(3*k))/(1 + x^(4*k))* kronecker(-12, k) = Sum_{k>0} (x^k + x^(3*k))/(1 + x^(2*k) + x^(4*k))* kronecker(8, k).
%o A128580 (PARI) {a(n)= if(n<0, 0, n=2*n+1; sumdiv(n, d, kronecker(-12, d)* kronecker(8, n/d)))}
%o A128580 (PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( eta(x+A)* eta(x^2+A)* eta(x^8+A)* eta(x^12+A)^3/ (eta(x^3+A)* eta(x^4+A)^2* eta(x^24+A)), n))}
%Y A128580 A128581(2n+1)= a(n). A128582(n)= -a(3n+2)/2. A113780(n)= a(12n).
%Y A128580 Adjacent sequences: A128577 A128578 A128579 this_sequence A128581 A128582 A128583
%Y A128580 Sequence in context: A074942 A043754 A094022 this_sequence A129402 A134177 A104405
%K A128580 sign
%O A128580 0,3
%A A128580 Michael Somos, Mar 11 2007
%I A129402
%S A129402 1,1,2,2,1,2,0,2,0,0,2,0,3,1,2,2,2,4,0,0,0,0,2,0,3,0,2,4,0,2,0,2,0,0,0,
%T A129402 0,2,3,4,2,1,2,0,2,0,0,2,0,2,2,2,2,4,2,0,0,0,0,0,0,3,0,4,2,0,2,0,2,0,0,
%U A129402 0,0,4,3,2,2,0,4,0,2,0,0,4,0,1,0,2,6,2,2,0,0,0,0,2,0,2,0,2,2,0,4,0,0,0
%N A129402 Expansion of phi(q^3)* psi(q^4) +q* phi(q)* psi(q^12) in powers of q where phi(), psi() are Ramanujan theta functions.
%D A129402 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 83, Eq. (32.57).
%F A129402 Euler transform of period 24 sequence [ 1, 1, 0, -2, 1, -1, 1, -1, 0, 1, 1, -2, 1, 1, 0, -1, 1, -1, 1, -2, 0, 1, 1, -2, ...].
%F A129402 a(n)=b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1, 5, 7, 11 (mod 24), b(p^e) = (1+(-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
%F A129402 a(12n+6)= a(12n+8)= a(12n+9)= a(12n+11)= 0. a(3n+1)= a(n).
%o A129402 (PARI) {a(n)= if(n<0, 0, n=2*n+1; sumdiv(n, d, kronecker(-6, d)))}
%o A129402 (PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( eta(x^3+A)* eta(x^4+A)^3* eta(x^6+A)* eta(x^24+A)/ (eta(x+A)* eta(x^8+A)* eta(x^12+A)^2), n))}
%Y A129402 A000377(2n+1)= a(n). A128582(n)= a(3n+2)/2. A113780(n)= a(12n).
%Y A129402 Adjacent sequences: A129399 A129400 A129401 this_sequence A129403 A129404 A129405
%Y A129402 Sequence in context: A043754 A094022 A128580 this_sequence A134177 A104405 A089077
%K A129402 nonn
%O A129402 0,3
%A A129402 Michael Somos, Apr 13 2007
%I A134177
%S A134177 1,1,2,2,1,2,0,2,0,0,2,0,3,1,2,2,2,4,0,0,0,0,2,0,3,0,2,4,0,2,0,2,0,0,0,
%T A134177 0,2,3,4,2,1,2,0,2,0,0,2,0,2,2,2,2,4,2,0,0,0,0,0,0,3,0,4,2,0,2,0,2,0,0,
%U A134177 0,0,4,3,2,2,0,4,0,2,0,0,4,0,1,0,2,6,2,2,0,0,0,0,2,0,2,0,2,2,0,4,0,0,0
%V A134177 1,1,-2,-2,1,2,0,-2,0,0,-2,0,3,1,-2,-2,2,4,0,0,0,0,-2,0,3,0,-2,-4,0,2,0,-2,0,0,0,0,2,3,
%W A134177 -4,-2,1,2,0,-2,0,0,-2,0,2,2,-2,-2,4,2,0,0,0,0,0,0,3,0,-4,-2,0,2,0,-2,0,0,0,0,4,3,-2,
%X A134177 -2,0,4,0,-2,0,0,-4,0,1,0,-2,-6,2,2,0,0,0,0,-2,0,2,0,-2,-2,0,4,0,0,0
%N A134177 Expansion of q^(-1/2) * eta(q^2)^4 * eta(q^3) * eta(q^8) * eta(q^12)^4 / ( eta(q) * eta(q^4)^3 * eta(q^6)^3 * eta(q^24) ) in powers of q.
%F A134177 Expansion of phi(-q^3) * psi(q^4) + q * phi(-q) * psi(q^12) in powers of q where phi(), psi() are Ramanujan theta functions.
%F A134177 a(n) = b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1, 11 (mod 24), b(p^e) = (e+1) * (-1)^e if p == 5, 7 (mod 24), b(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).
%F A134177 Euler transform of period 24 sequence [ 1, -3, 0, 0, 1, -1, 1, -1, 0, -3, 1, -2, 1, -3, 0, -1, 1, -1, 1, 0, 0, -3, 1, -2, ...].
%F A134177 G.f. is a period 1 Fourier series that satisfies f(-1 / (96 t)) = 96^(1/2) (t/i) f(t) where q = exp(2 pi i t).
%F A134177 a(12n+6) = a(12n+8) = a(12n+9) = a(12n+11) = 0.
%F A134177 G.f.: Sum_{k>=0} a(k) * x^(2*k+1) = Sum_{k>0} kronecker( -2, k) * (x^k - x^(3*k)) / (1 - x^(2*k) + x^(4*k)).
%F A134177 G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 + x^(6*k)) * (1 - x^(2*k) + x^(4*k))^2 / (1 - x^(4*k) + x^(8*k)).
%e A134177 q + q^3 - 2*q^5 - 2*q^7 + q^9 + 2*q^11 - 2*q^15 - 2*q^21 + 3*q^25 + q^27 + ...
%o A134177 (PARI) {a(n) = if( n<0, 0, n = 2*n+1; sumdiv(n, d, kronecker( 12, d)* kronecker( -2, n/d)))}
%o A134177 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^3 + A) * eta(x^8 + A) * eta(x^12 + A)^4 / ( eta(x + A) * eta(x^4 + A)^3 * eta(x^6 + A)^3* eta(x^24 + A) ), n))}
%o A134177 (PARI) {a(n) = local(A, p, e); if( n<0, 0, n = 2*n+1; A = factor(n); prod( k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 0, if(p==3, 1, if(p%24>12, !(e%2), (e+1)*kronecker(3,p)^e))))))}
%Y A134177 (-1)^n * A128580(n) = a(n). A113780(n)= a(12n).
%Y A134177 Adjacent sequences: A134174 A134175 A134176 this_sequence A134178 A134179 A134180
%Y A134177 Sequence in context: A094022 A128580 A129402 this_sequence A104405 A089077 A130071
%K A134177 sign
%O A134177 0,3
%A A134177 Michael Somos, Oct 11 2007
%I A104405
%S A104405 1,1,0,0,1,2,2,1,2,0,2,0,0,2,2,0,2,1,2,1,1,1,1,1,0,1,1,0,1,0,1,2,1,0,1,
%T A104405 0,1,1,1,2,2,0,0,2,0,1,2,1,0,0,2,0,0,1,2,0,0,1,0,0,2,2,2,1,2,0,0
%N A104405 A084203 (the cube root of A083953) read mod 3.
%H A104405 N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%Y A104405 Adjacent sequences: A104402 A104403 A104404 this_sequence A104406 A104407 A104408
%Y A104405 Sequence in context: A128580 A129402 A134177 this_sequence A089077 A130071 A038540
%K A104405 nonn
%O A104405 0,6
%A A104405 njas and Nadia Heninger *nadiah(AT)research.att.com), Jul 19 2005
%I A089077
%S A089077 0,0,1,0,0,1,0,1,1,2,2,1,2,0,2,2,0,1,2,1,4,1,3,4,1,5,5,4,8,1,8,4,4,7,
%T A089077 4,7,11,3,14,5,10,11,1,15,16,11,26,0,25,15,10,25,15,24,40,9,49,16,33,41,7,
%U A089077 50,57,34,90,6,83,56,27,89,62,82,145,26,171,63,108,146,37,172,209,109,316
%V A089077 0,0,1,0,0,-1,0,-1,-1,-2,-2,-1,-2,0,-2,2,0,1,2,1,4,-1,3,-4,-1,-5,-5,-4,-8,-1,-8,4,-4,7,
%W A089077 4,7,11,3,14,-5,10,-11,-1,-15,-16,-11,-26,0,-25,15,-10,25,15,24,40,9,49,-16,33,-41,-7,
%X A089077 -50,-57,-34,-90,6,-83,56,-27,89,62,82,145,26,171,-63,108,-146,-37,-172,-209,-109,-316
%N A089077 Alternating 4-Bonacci and theta1 Pisot 4 X 4 matrices in a recursion Adamson matrix procedure.
%C A089077 In 2d an odd even version plotted spirals outward.
%F A089077 q=1 m0={{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 0, 0, -q}} m1={{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 1, 1, q}} a(n) = If[Mod[n, 2]==0, m[n-1].m0, m[n-1].m1][[4, 4]]
%t A089077 (* Adamson's matrix functions alternating x^4-x^3-x^2-x-1 Pisot*) (* and x^4-x^3-1 minimal Pisot theta1*) digits=200 Solve[x^4-x^3-1==0, x] k=theta1 real root q=N[k-1/k^3, 20] m0={{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 0, 0, -q}} NSolve[x^4-x^3-x^2-x-1==0, x] k1=1.9275619754829254 q1=k1^2-k1-1/k1-1/k1^2 m1={{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 1, 1, q}} m[n_Integer?Positive] := If[Mod[n, 2]==0, m[n-1].m0, m[n-1].m1] m[0] ={{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}} a=Table[Floor[m[n][[4, 4]]], {n, 1, digits}]
%Y A089077 Adjacent sequences: A089074 A089075 A089076 this_sequence A089078 A089079 A089080
%Y A089077 Sequence in context: A129402 A134177 A104405 this_sequence A130071 A038540 A084348
%K A089077 sign,uned
%O A089077 1,10
%A A089077 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 04 2003
%I A130071
%S A130071 2,2,1,2,0,3,2,1,0,4,2,0,0,0,9,2,1,3,0,0,7,2,0,0,0,0,0,15,2,1,0,4,0,0,0,
%T A130071 12,2,0,3,0,0,0,0,0,18,2,1,0,0,9,0,0,0,0,17
%N A130071 Triangle, A007444(k) in each column interspersed with k zeros.
%C A130071 Row sums = the primes. T(n,k) = 0 if k does not divide n. If k divides n, extract A007444(k) which become the nonzero terms of row n, sum = n-th prime. Example: The factors of 6 are (1, 2, 3, and 6) = k's for A007444(k) = (2 + 1 + 3 + 7) = p(6) = 13. A007444 = the Moebius transform of the primes, (2, 1, 3, 4, 9, 7, 15, 12,...), as the right diagonal of A130071.
%F A130071 Given the Moebius transform of the primes, A007444: (2, 1, 3, 4, 9, 7, 15,...), the k-th term (k= 1,2,3,...) of this sequence generates the k-th column of A130071, interspersed with (k-1) zeros.
%e A130071 First few rows of the triangle are:
%e A130071 2;
%e A130071 2, 1;
%e A130071 2, 0, 3;
%e A130071 2, 1, 0, 4;
%e A130071 2, 0, 0, 0, 9;
%e A130071 2, 1, 3, 0, 0, 7;
%e A130071 2, 0, 0, 0, 0, 0, 15;
%e A130071 2, 1, 0, 4, 0, 0, 0, 12;
%e A130071 2, 0, 3, 0, 0, 0, 0, 0, 18;
%e A130071 2, 1, 0, 0, 9, 0, 0, 0, 0, 17;
%e A130071 ...
%Y A130071 Cf. A130070, A007444, A054525, A000040.
%Y A130071 Adjacent sequences: A130068 A130069 A130070 this_sequence A130072 A130073 A130074
%Y A130071 Sequence in context: A134177 A104405 A089077 this_sequence A038540 A084348 A085906
%K A130071 nonn,tabl
%O A130071 1,1
%A A130071 Gary W. Adamson (qntmpkt(AT)yahoo.com), May 05 2007
%I A038540
%S A038540 1,1,2,2,1,2,0,3,5,1,0,4,2,0,2,5,0,5,0,2,0,0,0,6,2,2,10,0,0,2,2,7,0,
%T A038540 0,0,10,2,0,4,3,2,0,2,0,5,0,0,10,1,2,0,4,2,10,0,0,0,0,0,4,0,2,0,11,
%U A038540 2,0,2,0,0,0,2,15,0,2,4,0,0,4,2,5,20,2,2,0,0,2,0,0,2,5,0,0,4,0,0,14
%N A038540 Number of modules with n elements over ring Z[ sqrt(10) ].
%D A038540 P. Samuel, Theorie algebrique des nombres, Hermann Editeurs.
%D A038540 D. Zagier, Zetafunktionen und quadratische Koerper, Springer-Verlag.
%F A038540 Define chi(n)=0 if n=2; legendre(10, n) if n is odd prime; chi(p1)^e1 * ... * chi(pk)^ek if n = p1^e1 * ... * pk^ek; Dirichlet g.f.: sum( a(n) / n^s, n=1..inf) = prod ( Z(ms), m=1..inf)
%Y A038540 Adjacent sequences: A038537 A038538 A038539 this_sequence A038541 A038542 A038543
%Y A038540 Sequence in context: A104405 A089077 A130071 this_sequence A084348 A085906 A090406
%K A038540 nonn,mult
%O A038540 1,3
%A A038540 Paolo Dominici (pl.dm(AT)libero.it)
%I A084348
%S A084348 0,0,1,2,2,1,2,1,0,1,2,0,1,0,1,2,5,4,5,2,1,2,5,2,2,4,4,1,2,5,0,1,6,5,4,
%T A084348 1,2,5,7,2,2,4,2,8,1,2,5,6,5,6,7,0,1,0,1,2,5,5,10,7,10,5,8,7,5,1,2,5,4,
%U A084348 5,2,1,8,5,10,5,8,1,2,5,3,0,1,7,11,11,9,0,1,0,1,2,5,2,9,4,11,8,9,12,9,2
%N A084348 Triangle in which row n gives periodic part of a certain map.
%C A084348 Let r(k,n)=floor(e*k!)-n*floor(e*k!/n) then for any n integer>0, sequence r(k,n) is n-periodic. Sequence gives periods of r(k,n) for fixed n.
%e A084348 If n=7, r(k,7) is sequence 2, 5, 2, 2, 4, 4, 1, 2, 5, 2, 2, 4, 4, 1, 2, 5, 2, 2, 4, 4, 1, 2, 5, 2, 2, 4, 4, 1, 2, 5...... 7-periodic with period: (2, 5, 2, 2, 4, 4, 1,)
%Y A084348 Cf. A084351.
%Y A084348 Adjacent sequences: A084345 A084346 A084347 this_sequence A084349 A084350 A084351
%Y A084348 Sequence in context: A089077 A130071 A038540 this_sequence A085906 A090406 A137454
%K A084348 nonn,tabl
%O A084348 1,4
%A A084348 Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 22 2003
%I A085906
%S A085906 1,1,2,2,1,2,1,0,3,1,1,4,1,1,2,0,1,3,1,2,2,1,1,0,0,1,0,2,1,2,1,0,2,1,1,
%T A085906 6,1,1,2,0,1,2,1,2,3,1,1,0,0,0,2,2,1,0,1,0,2,1,1,4,1,1,3,0,1,2,1,2,2,1,
%U A085906 1,0,1,1,0,2,1,2,1,0,0,1,1,4,1,1,2,0,1,3,1,2,2,1,1,0,1,0,3,0
%V A085906 1,1,2,-2,-1,2,-1,0,-3,-1,-1,-4,-1,-1,-2,0,-1,-3,-1,2,-2,-1,-1,0,0,-1,0,2,-1,-2,-1,0,
%W A085906 -2,-1,1,6,-1,-1,-2,0,-1,-2,-1,2,3,-1,-1,0,0,0,-2,2,-1,0,1,0,-2,-1,-1,4,-1,-1,3,0,1,-2,
%X A085906 -1,2,-2,1,-1,0,-1,-1,0,2,1,-2,-1,0,0,-1,-1,4,1,-1,-2,0,-1,3,1,2,-2,-1,1,0,-1,0,3,0
%N A085906 Ramanujan sum c_6(n).
%D A085906 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
%F A085906 a(n) = phi(n)*mu(n/gcd(n, 6)) / phi(n/gcd(n, 6)).
%o A085906 (PARI) a(n)=eulerphi(n)*moebius(n/gcd(n,6))/eulerphi(n/gcd(n,6))
%Y A085906 Cf. A085639.
%Y A085906 Adjacent sequences: A085903 A085904 A085905 this_sequence A085907 A085908 A085909
%Y A085906 Sequence in context: A130071 A038540 A084348 this_sequence A090406 A137454 A030613
%K A085906 sign,mult
%O A085906 1,3
%A A085906 Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 16 2003
%E A085906 More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 18 2003
%I A090406
%S A090406 2,2,1,2,1,1,0,1,1,2,1,1,0,1,1,2,1,1,0,1,1,1,0,0,0,1,1,2,1,1,0,0,0,1,1,
%T A090406 1,0,1,1,2,1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,2,1,1,0,0,0,1,1,1,0,1,1,2,
%U A090406 1,1,0,0,0,1,1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0,0,1,1,1,0,1,1,2,1,1
%N A090406 PrimePi[n+3]-PrimePi[n].
%H A090406 Eric Weisstein's World of Mathematics, Hardy-Littlewood Conjectures
%Y A090406 Cf. A080545, A090405.
%Y A090406 Adjacent sequences: A090403 A090404 A090405 this_sequence A090407 A090408 A090409
%Y A090406 Sequence in context: A038540 A084348 A085906 this_sequence A137454 A030613 A092984
%K A090406 nonn
%O A090406 1,1
%A A090406 Eric Weisstein (eric(AT)weisstein.com), Nov 29, 2003
%I A137454
%S A137454 1,1,1,1,1,1,1,1,1,2,2,1,2,1,1,1,0,1,5,4,1,3,1,1,1,3,5,2,3,9,6,1,4,1,1,
%T A137454 1,2,1,10,16,6,6,14,8,1,5,1,1,1,3,9,14,4,23,34,12,10,20,10,1,6,1,1,1,5,
%U A137454 8,10,38,45,10,44,60,20,15,27,12,1,7,1,1,1,1,11,38,42,23,101,105,20,75
%V A137454 1,-1,-1,1,-1,-1,-1,-1,1,2,2,-1,-2,-1,-1,1,0,1,5,4,-1,-3,-1,-1,1,-3,-5,-2,3,9,6,-1,-4,
%W A137454 -1,-1,1,2,1,-10,-16,-6,6,14,8,-1,-5,-1,-1,1,3,9,14,4,-23,-34,-12,10,20,10,-1,-6,-1,-1,
%X A137454 1,-5,-8,10,38,45,10,-44,-60,-20,15,27,12,-1,-7,-1,-1,1,-1,-11,-38,-42,23,101,105,20
%N A137454 A triangular sequence from coefficients of a three deep quadratic polynomial recursion: p(x,n)=x^2*p(x,n-1)+(-x-1)*p(x,n-2)+p(x,n-3).
%C A137454 Row sums are:
%C A137454 {1, -1, -3, 0, 5, 2, -8, -7, 11, 17, -12, -35, 6, 64, 17, -105, -75, 152, 197, -182, -424};
%C A137454 The row sums appear to be a saw tooth alternating sequence.
%C A137454 The sequence is designed so the first three terms are all ones
%C A137454 and it will have an odd number of coefficient elements.
%F A137454 p(x,-1)=0;p(x,0)=1;p(x,1)=x^2-x-2; p(x,n)=x^2*p(x,n-1)+(-x-1)*p(x,n-2)+p(x,n-3).
%e A137454 {1},
%e A137454 {-1, -1, 1},
%e A137454 {-1, -1, -1, -1, 1},
%e A137454 {2, 2, -1, -2, -1, -1, 1},
%e A137454 {0, 1, 5, 4, -1, -3, -1, -1, 1},
%e A137454 {-3, -5, -2, 3,9, 6, -1, -4, -1, -1, 1},
%e A137454 {2, 1, -10, -16, -6, 6,14, 8, -1, -5, -1, -1, 1},
%e A137454 {3, 9,14, 4, -23, -34, -12, 10, 20, 10, -1, -6, -1, -1, 1},
%e A137454 {-5, -8, 10, 38, 45, 10, -44, -60, -20, 15,27, 12, -1, -7, -1, -1, 1},
%e A137454 {-1, -11, -38, -42, 23, 101, 105,20, -75, -95, -30, 21, 35, 14, -1, -8, -1, -1,1},
%e A137454 {8, 22, 11, -55, -144, -131, 45, 215, 205, 35, -118, -140, -42, 28, 44, 16, -1, -9, -1, -1, 1}
%t A137454 Clear[A, x, p] p[x, -1] = 0; p[x, 0] = 1; p[x, 1] = x^2 - x - 1; p[x_, n_] := x^2*p[x, n - 1] + (-x - 1)*p[x, n - 2] + p[x, n - 3]; Table[Expand[p[x, n]], {n, 0, 10}]; a1 = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a1]
%Y A137454 Cf. A084610.
%Y A137454 Adjacent sequences: A137451 A137452 A137453 this_sequence A137455 A137456 A137457
%Y A137454 Sequence in context: A084348 A085906 A090406 this_sequence A030613 A092984 A086600
%K A137454 nonn,tabl,uned,new
%O A137454 1,10
%A A137454 Roger L. Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Apr 18 2008
%I A030613
%S A030613 1,1,1,1,1,1,1,2,2,1,2,1,1,1,1,1,1,2,1,1,1,1,3,1,1,1,1,1,1,1,
%T A030613 1,1,1,1,1,1,1,1,3,1,1,2,2,1,1,1,1,2,1,1,1,1,2,1,1,1,1,1,1,1,
%U A030613 1,1,1,1,2,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1
%N A030613 Length of n-th run of digit 2 in A030604.
%Y A030613 Adjacent sequences: A030610 A030611 A030612 this_sequence A030614 A030615 A030616
%Y A030613 Sequence in context: A085906 A090406 A137454 this_sequence A092984 A086600 A025912
%K A030613 nonn
%O A030613 1,8
%A A030613 Clark Kimberling (ck6(AT)evansville.edu)
%I A092984
%S A092984 1,1,1,2,2,1,2,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,
%T A092984 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A092984 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A092984 Least k >1 such that n! + k is square-free.
%C A092984 Conjecture: There exists a finite k such that a(n) < k for all n. Subsidiary sequence: Indices of the first occurrence of n in this sequence. In case the conjecture is true, this sequence would be finite.
%C A092984 If a(n)=2 ==> n!+1 is divisible by a square (sequence A064237) - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 29 2004
%F A092984 A092983(n)-n!.
%e A092984 a(5) = 2 = 122-120.
%o A092984 (PARI) a(n)=for(i=1,n!,if(issquarefree(n!+i),return(i)))
%Y A092984 Cf. A092983.
%Y A092984 Cf. A064237.
%Y A092984 Adjacent sequences: A092981 A092982 A092983 this_sequence A092985 A092986 A092987
%Y A092984 Sequence in context: A090406 A137454 A030613 this_sequence A086600 A025912 A029441
%K A092984 nonn
%O A092984 1,4
%A A092984 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 28 2004
%E A092984 More terms from Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 29 2004
%E A092984 More terms from David Wasserman (dwasserm(AT)earthlink.net), Sep 27 2006
%I A086600
%S A086600 0,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,1,2,1,1,1,2,1,1,2,1,2,2,
%T A086600 1,1,1,2,2,1,2,2,3,2,2,1,2,1,2,2,2,1,1,2,1,2,2,3,2,1,1,2,2,3,1,2,1,2,1,
%U A086600 1,1,2,2,2,1,2,1,1,1,2,3,2,2,2,1,2,2,2,1,1,2,2,3,3,1,2,2,3,2,3,2,3,3,2
%N A086600 Number of primitive prime factors in Lucas(n).
%C A086600 A prime factor of Lucas(n) is called primitive if it does not divide Lucas(r) for any r < n. It can be shown that there is at least one primitive prime factor for n > 6. When n is prime, all the prime factors of Lucas(n) are primitive.
%H A086600 T. D. Noe, Table of n, a(n) for n=1..1000 (using Blair Kelly's data)
%H A086600 Blair Kelly, Fibonacci and Lucas Factorizations
%H A086600 Eric Weisstein's World of Mathematics, Lucas Number
%F A086600 a(n) = Sum{d|n and n/d odd} mu(n/d) A086600(d) -1 if 6|n and n/6 is a power of 2.
%e A086600 a(22) = 2 because Lucas(22) = 3*43*307 and neither 43 nor 307 divide a smaller Lucas number.
%t A086600 Lucas[n_] := Fibonacci[n+1] + Fibonacci[n-1]; pLst={}; Join[{0}, Table[f=Transpose[FactorInteger[Lucas[n]]][[1]]; f=Complement[f, pLst]; cnt=Length[f]; pLst=Union[pLst, f]; cnt, {n, 2, 150}]]
%Y A086600 Cf. A000204 (Lucas numbers), A086598 (number of distinct prime factors), A086599 (number of prime factors, counting multiplicity).
%Y A086600 Adjacent sequences: A086597 A086598 A086599 this_sequence A086601 A086602 A086603
%Y A086600 Sequence in context: A137454 A030613 A092984 this_sequence A025912 A029441 A109495
%K A086600 hard,nonn
%O A086600 1,22
%A A086600 T. D. Noe (noe(AT)sspectra.com), Jul 24 2003
%I A025912
%S A025912 1,0,0,0,0,0,0,1,0,1,1,0,0,0,1,0,1,1,1,1,1,1,0,1,1,1,1,
%T A025912 2,2,1,2,1,1,1,2,2,2,3,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,
%U A025912 4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,5,6,6,6,6,6,7,6,7
%N A025912 Expansion of 1/((1-x^7)(1-x^9)(1-x^10)).
%Y A025912 Adjacent sequences: A025909 A025910 A025911 this_sequence A025913 A025914 A025915
%Y A025912 Sequence in context: A030613 A092984 A086600 this_sequence A029441 A109495 A035214
%K A025912 nonn
%O A025912 0,28
%A A025912 njas
%I A029441
%S A029441 1,0,0,0,0,0,0,1,0,0,1,1,1,0,1,0,0,1,1,1,1,2,2,1,2,1,1,
%T A029441 1,2,2,2,3,3,3,3,3,3,2,3,3,4,4,5,5,5,5,5,5,5,5,6,6,7,7,
%U A029441 8,8,8,8,8,8,9,9,10,10,11,11,12,12,12,12,13,13,14,14,15
%N A029441 Expansion of 1/((1-x^7)(1-x^10)(1-x^11)(1-x^12)).
%Y A029441 Adjacent sequences: A029438 A029439 A029440 this_sequence A029442 A029443 A029444
%Y A029441 Sequence in context: A092984 A086600 A025912 this_sequence A109495 A035214 A071292
%K A029441 nonn
%O A029441 0,22
%A A029441 njas
%I A109495
%S A109495 1,2,2,1,2,1,1,1,2,5,2,1,5,5,1,1,2,1,1,2,1,2,8,8,2,1,8,15,8,1,2,8,8,2,
%T A109495 1,2,1,1,3,3,1,2,11,18,11,2,1,11,31,31,11,1,3,18,31,18,3,3,11,11,3,1,
%U A109495 2,1,1,4,6,4,1,2,14,32,32,14,2,1,14,53,80,53,14,1,4,32,80,80,32,4,6
%N A109495 Entries in 3-dimensional solid related to Prouhet-Tarry problem.
%C A109495 Entries of slices [2,k] in A109672, read by rows.
%C A109495 Slice [n,0] gives A046816, slice [0,k] gives A109649, slice [n,n] gives A109673, slice [n,1] gives A109390, slice [1,k] gives A109393.
%F A109495 Sum of terms in 2D slice [2, k] is 3^(2+k).
%e A109495 Slice [2,0]:
%e A109495 .... 1 ....
%e A109495 ... 2 2 ...
%e A109495 .. 1 2 1 ...
%e A109495 Slice [2,1]:
%e A109495 .... 1 1 ....
%e A109495 ... 2 5 2 ...
%e A109495 .. 1 5 5 1 ...
%e A109495 ... 1 2 1 ...
%e A109495 Slice [2,2]:
%e A109495 .... 1 2 1 ....
%e A109495 ... 2 8 8 2 ...
%e A109495 .. 1 8 15 8 1 ...
%e A109495 ... 2 8 8 2 ...
%e A109495 .... 1 2 1 ....
%e A109495 Slice [2,3]:
%e A109495 .... 1 3 3 1 ....
%e A109495 ... 2 11 18 11 2 ...
%e A109495 .. 1 11 31 31 11 1 ...
%e A109495 ... 3 18 31 18 3 ....
%e A109495 .... 3 11 11 3 .....
%e A109495 ..... 1 2 1 ......
%e A109495 Slice [2,4]:
%e A109495 .... 1 4 6 4 1 ...
%e A109495 ... 2 14 32 32 14 2 ...
%e A109495 .. 1 14 53 80 53 14 1 ...
%e A109495 ... 4 32 80 80 32 4 ....
%e A109495 .... 6 32 53 32 6 .....
%e A109495 ..... 4 14 14 4 .....;
%e A109495 ...... 1 2 1 ......;
%e A109495 Slice [2,5]:
%e A109495 .... 1 5 10 10 5 1 ...
%e A109495 ... 2 17 50 70 50 17 2 ...
%e A109495 .. 1 17 81 165 165 81 17 1 ...
%e A109495 ... 5 50 165 240 165 50 5 ....
%e A109495 .... 10 70 165 165 70 10 .....
%e A109495 ..... 10 50 81 50 10 ......
%e A109495 ...... 5 17 17 5 ......
%e A109495 ....... 1 2 1 .......
%Y A109495 Cf. A000001.
%Y A109495 Adjacent sequences: A109492 A109493 A109494 this_sequence A109496 A109497 A109498
%Y A109495 Sequence in context: A086600 A025912 A029441 this_sequence A035214 A071292 A088569
%K A109495 nonn,tabf,easy
%O A109495 0,2
%A A109495 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 29 2005
%I A035214
%S A035214 2,2,1,2,1,1,2,1,1,1,2,1,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1,1,2,1,1,1,1,
%T A035214 1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,
%U A035214 2,1,1,1,1,1,1,1,1,1,1,1,1
%N A035214 2 followed by a run of k 1's.
%F A035214 a(n) = 2 if n is a triangular number, otherwise 1.
%F A035214 a(n) = floor((3-cos(Pi*sqrt(8*n+1)))/2) - Carl R. White (oeisfan(AT)cyreksoft.yorks.com), Mar 18 2006
%Y A035214 Cf. A035253, A035254.
%Y A035214 A010054(n) + 1.
%Y A035214 Adjacent sequences: A035211 A035212 A035213 this_sequence A035215 A035216 A035217
%Y A035214 Sequence in context: A025912 A029441 A109495 this_sequence A071292 A088569 A001285
%K A035214 nonn
%O A035214 0,1
%A A035214 njas
%I A071292
%S A071292 1,1,1,1,1,1,1,2,2,1,2,1,1,2,1,1,2,1,2,1,2,1,2,1,2,1,2,1,1,2,2,1,2,1,
%T A071292 2,1,2,1,2,2,1,2,1,2,2,1,2,1,2,2,1,2,1,2,2,1,2,2,1,2,1,2,2,1,2,2,1,2,
%U A071292 2,2,1,2,2,1,2,2,1,2,2,1,2,2,2,1,2,2,1,2,2,1,2,2,2,1,2,2,1,2,2,2
%N A071292 Call f(n) the sum of the first n primes then a(n) is the number of squares between f(n) and f(n+1).
%C A071292 a(n)>=1.
%H A071292 Internet Math. Olympiad, February: Problem 1
%t A071292 squareQ[n_] := IntegerQ[Sqrt[n]] f[n_] := Sum[Prime[i], {i, 1, n}] a[n_] := Length[Select[Table[i, {i, a[n], a[n + 1]}], squareQ]]
%Y A071292 Adjacent sequences: A071289 A071290 A071291 this_sequence A071293 A071294 A071295
%Y A071292 Sequence in context: A029441 A109495 A035214 this_sequence A088569 A001285 A088424
%K A071292 nonn
%O A071292 1,8
%A A071292 Santi Spadaro (spados(AT)katamail.com), Jun 11 2002
%I A088569
%S A088569 1,1,2,2,1,2,1,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,1,1,2,2,1,2,2,1,2,1,1,
%T A088569 2,1,1,2,2,1,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,1,2,2,1,2,2,
%U A088569 1,2,1,1,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2
%N A088569 Anti-Kolakoski sequence (sequence of length runs never coincides with the sequence itself).
%C A088569 Unique infinite word defined on alphabet {1,2} satisfying: a(1)=1, if a(n)=2 length of n-th run is 1, if a(n)=1 length of n-th run is 2. Kolakoski sequence satisfies the opposite definition : K(1)=1, if K(n)=2 length of n-th run is 2, if K(n)=1 length of n-th run is 1.
%F A088569 a(n)=3-A000002(n+1)=A049705(n+1)
%e A088569 a(1)=1 hence first run must have length 2 and necessarily a(2)=1. Now second run must have also length 2 and therefore a(3)=a(4)=2.
%Y A088569 Adjacent sequences: A088566 A088567 A088568 this_sequence A088570 A088571 A088572
%Y A088569 Sequence in context: A109495 A035214 A071292 this_sequence A001285 A088424 A097456
%K A088569 nonn
%O A088569 1,3
%A A088569 Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 17 2003
%I A001285 M0193 N0071
%S A001285 1,2,2,1,2,1,1,2,2,1,1,2,1,2,2,1,2,1,1,2,1,2,2,1,1,2,2,1,2,1,1,2,2,1,
%T A001285 1,2,1,2,2,1,1,2,2,1,2,1,1,2,1,2,2,1,2,1,1,2,2,1,1,2,1,2,2,1,2,1,1,2,
%U A001285 1,2,2,1,1,2,2,1,2,1,1,2,1,2,2,1,2,1,1,2,2,1,1,2,1,2,2,1,1,2,2,1,2,1
%N A001285 Thue-Morse sequence: let A_k denote the first 2^k terms; then A_0 = 1, and for k >= 0, A_{k+1} = A_k B_k, where B_k is obtained from A_k by interchanging 1's and 2's.
%C A001285 Or, follow a(0), .., a(2^k-1) by its complement.
%D A001285 J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 15.
%D A001285 F. Axel et al., Vibrational modes in a one dimensional "quasi-alloy": the Morse case, J. de Physique, Colloq. C3, Supp. to No. 7, Vol. 47 (Jul 1986), pp. C3-181-C3-186; see Eq. (10).
%D A001285 F. Dejean, Sur un theoreme de Thue. J. Combinatorial Theory Ser. A 13 (1972), 90-99.
%D A001285 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A001285 W. H. Gottschalk and G. A. Hedlund, Topological Dynamics. American Mathematical Society, Colloquium Publications, Vol. 36, Providence, RI, 1955, p. 105.
%D A001285 G. A. Hedlund, Remarks on the work of Axel Thue on sequences, Nordisk Mat. Tid., 15 (1967), 148-150.
%D A001285 A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schroedinger operators, Commun. Math. Phys. 174 (1995), 149-159.
%D A001285 M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 23.
%D A001285 M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc., 22 (1921), 84-100.
%D A001285 A. Salomaa, Jewels of Formal Language Theory. Computer Science Press, Rockville, MD, 1981, p. 6.
%H A001285 T. D. Noe, Table of n, a(n) for n = 0..1023
%H A001285 J.-P. Allouche and J. O. Shallit, The Ubiquitous Prouhet-Thue-Morse Sequence, in C. Ding. T. Helleseth, and H. Niederreiter, eds., Sequences and Their Applications: Proceedings of SETA '98, Springer-Verlag, 1999, pp. 1-16.
%H A001285 Michael Gilleland, Some Self-Similar Integer Sequences
%H A001285 S. Wolfram, Source for short Thue-Morse generating code
%H A001285 Index entries for "core" sequences
%F A001285 a(2n)=a(n), a(2n+1)=3-a(n), a(0)=1. Also, a(k+2^m)=3-a(k) if 0<=k<2^m.
%F A001285 a(n) = 2-A010059(n) = 1/2*(3-(-1)^A000120(n)). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 20 2003
%F A001285 a(n)=sum(k=0, n, binomial(n, k){mod 2}) {mod 3}=A001316(n) {mod 3} - Benoit Cloitre (benoit7848c(AT)orange.fr), May 09 2004
%p A001285 A001285 := proc(n) option remember; if n=0 then 1 elif n mod 2 = 0 then A001285(n/2) else 3-A001285((n-1)/2); fi; end;
%p A001285 s := proc(k) local i, ans; ans := [ 1,2 ]; for i from 0 to k do ans := [ op(ans),op(map(n->if n=1 then 2 else 1 fi, ans)) ] od; RETURN(ans); end; t1 := s(6); A001285 := n->t1[n]; # s(k) gives first 2^(k+2) terms
%t A001285 Nest[ Function[l, {Flatten[(l /. {2 -> {2, 1}, 1 -> {1, 2}})]}], {1}, 7] (from Robert G. Wilson v Feb 26 2005)
%o A001285 (PARI) a(n)=1+subst(Pol(binary(n)),x,1)%2
%o A001285 (PARI) a(n)=sum(k=0,n,binomial(n,k)%2)%3
%Y A001285 Cf. A010060 (for 0,1 version), A003159. A001285(n)=1+A010060(n).
%Y A001285 A026465 gives run lengths.
%Y A001285 Cf. A010059 (1, 0 version).
%Y A001285 Adjacent sequences: A001282 A001283 A001284 this_sequence A001286 A001287 A001288
%Y A001285 Sequence in context: A035214 A071292 A088569 this_sequence A088424 A097456 A087775
%K A001285 nonn,easy,core,nice
%O A001285 0,2
%A A001285 njas
%I A088424
%S A088424 1,2,2,1,2,1,1,2,2,1,1,3,1,2,4,1,2,1,1,2,1,1,2,4,1,1,2,1,2,2,1,1,2,1,2,
%T A088424 1,1,2,1,1,1,3,1,2,6,1,2,2,1,2,1,1,1,1,1,1,3,1,2,1,1,2,2,1,1,1,1,2,2,1,
%U A088424 1,2,1,1,5,1,2,1,1,2,1,1,2,3,1,1,1,1,2,3,1,2,2,1,1,1,1,1,4,1
%N A088424 Number of primes in arithmetic progression starting with 13 and with d=2n.
%C A088424 Arithmetic progression is stopped when next term is not prime. E.g. for n=15, a=4, that is 13,43,73,103 are prime, while next term, 133, is not prime.
%t A088424 bb={}; Do[s=1; Do[If[PrimeQ[13+k*d], s=s+1, bb={bb, s}; Break[]], {k, 10}], {d, 2, 200, 2}]; Flatten[bb]
%Y A088424 Cf. A088420, A088421, A088422, A088423, A088425, A088426, A088427, A088428, A088429.
%Y A088424 Adjacent sequences: A088421 A088422 A088423 this_sequence A088425 A088426 A088427
%Y A088424 Sequence in context: A071292 A088569 A001285 this_sequence A097456 A087775 A089955
%K A088424 easy,nonn
%O A088424 1,2
%A A088424 Zak Seidov (zakseidov(AT)yahoo.com), Sep 29 2003
%I A097456
%S A097456 1,0,1,0,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,1,2,1,1,2,2,2,2,
%T A097456 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%U A097456 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,2,2,2,3,2,2,2,2,2,2
%N A097456 Integer part of the ratio (number of composites <=n) / (number of primes <=n).
%F A097456 pi(n) = number of primes <= n. Ratio = (n - pi(n))/pi(n) = n/pi(n) - 1. Output floor(Ratio)
%o A097456 (PARI) compsp1(n) = { for(x=2,n, y=floor(x/pi(x)-1); print1(y",") ) }
%Y A097456 Adjacent sequences: A097453 A097454 A097455 this_sequence A097457 A097458 A097459
%Y A097456 Sequence in context: A088569 A001285 A088424 this_sequence A087775 A089955 A046816
%K A097456 nonn
%O A097456 2,26
%A A097456 Cino Hilliard (hillcino368(AT)gmail.com), Aug 23 2004
%I A087775
%S A087775 1,2,2,1,2,1,1,2,4,1,2,7,1,2,10,1,2,13,1,2,16,1,2,19,1,2,22,1,2,25,1,2,28,1,
%T A087775 2,31,1,2,34,1,2,37,1,2,40,1,2,43,1,2,46,1,2,49,1,2,52,1,2,55,1,2,58,1,2,61,
%U A087775 1,2,64,1,2,67,1,2,70,1,2,73,1,2,76,1,2,79,1,2,82,1,2,85,1,2,88,1,2,91,1,2
%V A087775 1,2,2,1,2,-1,1,2,-4,1,2,-7,1,2,-10,1,2,-13,1,2,-16,1,2,-19,1,2,-22,1,2,-25,1,2,-28,1,
%W A087775 2,-31,1,2,-34,1,2,-37,1,2,-40,1,2,-43,1,2,-46,1,2,-49,1,2,-52,1,2,-55,1,2,-58,1,2,-61,
%X A087775 1,2,-64,1,2,-67,1,2,-70,1,2,-73,1,2,-76,1,2,-79,1,2,-82,1,2,-85,1,2,-88,1,2,-91,1,2
%N A087775 a[1] = 1, a[2] = 2, a[3] = 2; a[n] = 3*a[abs[a[n-2]]] - 3*a[n-abs[a[n-2]]] + a[n-3].
%C A087775 A recursive sequence based on A087774 and the Mallows A005229 sequence. A tick-tock sequence where the form is: 1,2,a(n-3)-3
%t A087775 hg[n_Integer?Positive] := hg[n] =3*hg[Abs[hg[n-2]]] -3*hg[n-Abs[hg[n-2]]]+hg[n-3] hg[1] =1 hg[2] =2 hg[3]=2 digits=200 a=Table[hg[n], {n, 1, digits}]
%Y A087775 Cf. A005229.
%Y A087775 Adjacent sequences: A087772 A087773 A087774 this_sequence A087776 A087777 A087778
%Y A087775 Sequence in context: A001285 A088424 A097456 this_sequence A089955 A046816 A138328
%K A087775 sign
%O A087775 1,2
%A A087775 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Oct 04 2003
%I A089955
%S A089955 0,0,0,0,0,0,2,2,1,2,1,1,3,1,1,1,2,3,3,1,1,3,3,2,3,7,3,5,4,3,5,5,5,4,3,
%T A089955 5,5,4,4,10,7,3,4,4,7,7,7,3,2,7,5,6,8,6,9,9,7,7,5,8,6,8,7,8,6,8,10,8,3,
%U A089955 7,10,6,10,8,9,6,8,7,12,11,9,11,7,8,11,7,11,10,6,9,8,9,13,11,7,13,14,11
%N A089955 Number of embedded Fibonacci numbers within the n-th Fibonacci number.
%H A089955 Gil Broussard, The Embedded Fibonacci Sequence.
%e A089955 a(26)=7 because Fib(26)=121393, and 7 Fibonacci numbers appear within 121393: 1 appears twice, 3 appears twice, and 2, 13, and 21 each appear once.
%Y A089955 Cf. A000045.
%Y A089955 Adjacent sequences: A089952 A089953 A089954 this_sequence A089956 A089957 A089958
%Y A089955 Sequence in context: A088424 A097456 A087775 this_sequence A046816 A138328 A137264
%K A089955 nonn,base
%O A089955 1,7
%A A089955 Gil Broussard (kikiriki(AT)mindspring.com), Jan 12 2004
%I A046816
%S A046816 1,1,1,1,1,2,2,1,2,1,1,3,3,3,6,3,1,3,3,1,1,4,4,6,12,6,4,12,12,4,1,4,6,
%T A046816 4,1,1,5,5,10,20,10,10,30,30,10,5,20,30,20,5,1,5,10,10,5,1,1,6,6,15,30,
%U A046816 15,20,60,60,20,15,60,90,60,15,6,30,60,60,30,6,1,6,15,20,15,6,1
%N A046816 Entries in 3-dimensional version of Pascal triangle: trinomial coefficients.
%C A046816 Greatest numbers in each 2D triangle form A022916 (multinomial coefficient n!/([n/3]![(n+1)/3]![(n+2)/3]!).) 2D triangle sums are powers of 3. - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Aug 15 2004
%F A046816 Coefficients of x, y, z in (x+y+z)^n: a(i+1, k, j) = a(i, k, j)+a(i, j, k-1)+a(i, j-1, k-1), a(i, j, -1) := 0, ...; a(0, 0, 0)=1.
%e A046816 ... 1 .... Here is the third slice of the pyramid
%e A046816 .. 3 3
%e A046816 . 3 6 3
%e A046816 .1 3 3 1
%Y A046816 Cf. A007318.
%Y A046816 Entry [3,2] in each slice gives A002378, entry [4,3] in each slice gives A027480, entry [5,2] in each slice gives A033488, entry [5,3] in each slice gives A033487.
%Y A046816 Adjacent sequences: A046813 A046814 A046815 this_sequence A046817 A046818 A046819
%Y A046816 Sequence in context: A097456 A087775 A089955 this_sequence A138328 A137264 A078614
%K A046816 nonn,tabf,easy
%O A046816 0,6
%A A046816 Lior Manor (lior.manor(AT)gmail.com)
%I A138328
%S A138328 0,0,1,0,1,0,1,1,0,1,0,1,1,0,1,0,1,1,2,2,1,2,1,2,0,0,1,2,2,1,2,2,2,3,0,
%T A138328 0,2,1,2,2,0,3,3,3,0,1,2,4,0,2,0,4,2,2,5,4,2,3,1,2,4,2,3,0,3,1,5,1,6,3,
%U A138328 4,4,4,2,2,3,3,5,3,3,0,4,1,3,5,4,5,4,2,2,4,6,3,1,2,4,2,4,4,0,5,6,5,7,2
%N A138328 Number of ways in which n can be represented as p+g, where p is a prime and g is a primitive root of p.
%H A138328 T. D. Noe, Table of n, a(n) for n=1..10000
%Y A138328 Cf. A060749 (primitive roots), A138325 (least p such that n=p+g).
%Y A138328 Adjacent sequences: A138325 A138326 A138327 this_sequence A138329 A138330 A138331
%Y A138328 Sequence in context: A087775 A089955 A046816 this_sequence A137264 A078614 A026607
%K A138328 nonn
%O A138328 1,19
%A A138328 T. D. Noe (noe(AT)sspectra.com), Mar 14 2008
%I A137264
%S A137264 1,2,2,1,2,1,2,1,0,2,0,1,2,1,0,0,2,0,1,2,0,1,0,2,1,2,1,2,1,2,1,0,2,1,2,
%T A137264 0,0,1,0,0,2,1,2,1,2,0,0,1,2,1,0,2,1,0,0,0,2,0,1,2,1,2,1,2,1,2,0,1,2,1,
%U A137264 0,2,0,0,1,0,2,1,2,1,2,1,2,0,1,0,2,1,2,1,0,2,1,2,1,0,0,2,0,0,1,0
%N A137264 Prime number gaps read modulo 3.
%C A137264 Conjecture: The only digit that is repeated in the sequence is 0 except for n=2 and n=3 where 2 repeats. So 1 may be followed by 2 or 0; 2 may be followed by 1 or 0; 0 may be followed by 0 or 1 or 2. this has been confirmed for the first million prime gaps.
%t A137264 n=1000;(*The length of the list*) Mod[Differences[Table[Prime[i], {i, n}]], 3]
%Y A137264 Cf. A001223.
%Y A137264 Adjacent sequences: A137261 A137262 A137263 this_sequence A137265 A137266 A137267
%Y A137264 Sequence in context: A089955 A046816 A138328 this_sequence A078614 A026607 A052005
%K A137264 easy,nonn
%O A137264 1,2
%A A137264 Noel H. Patson (n.patson(AT)cqu.edu.au), Mar 12 2008
%I A078614
%S A078614 2,2,1,2,1,2,1,2,1,1,2,1,2,1,2,1,2,1,1,2,1,2,1,2,1,2,1,1,2,1,2,1,2,1,1,
%T A078614 2,1,2,1,2,1,2,1,1,2,1,2,1,2,1,2,1,1,2,1,2,1,2,1,2
%N A078614 Differences of A072633.
%e A078614 a(1)=2, as A072633(2)=4 and A072633(1)=2
%o A078614 (PARI) pu(m,n)=local(s); s=0; for (i=1,m,s=s+i^n); s ox=1; for (k=1,60,x=1; while (pu(x,k)<(x+1)^k,x++); print1((x-ox)","); ox=x)
%Y A078614 Cf. A072633.
%Y A078614 Adjacent sequences: A078611 A078612 A078613 this_sequence A078615 A078616 A078617
%Y A078614 Sequence in context: A046816 A138328 A137264 this_sequence A026607 A052005 A138702
%K A078614 nonn
%O A078614 1,1
%A A078614 Jon Perry (perry(AT)globalnet.co.uk), Dec 10 2002
%I A026607
%S A026607 1,2,2,1,2,1,2,1,2,1,1,2,2,1,1,2,2,1,2,1,2,1,1,2,2,1,1,2,2,1,
%T A026607 1,2,2,1,2,1,2,1,1,2,2,1,1,2,2,1,2,1,2,1,2,1,1,2,2,1,2,1,1,2,
%U A026607 2,1,1,2,2,1,1,2,2,1,2,1,2,1,1,2,2,1,1,2,2,1,2,1,2,1,2,1,1,2
%N A026607 Delete all 2's from A026600 and then replace each 3 by 2.
%Y A026607 Adjacent sequences: A026604 A026605 A026606 this_sequence A026608 A026609 A026610
%Y A026607 Sequence in context: A138328 A137264 A078614 this_sequence A052005 A138702 A112104
%K A026607 nonn
%O A026607 1,2
%A A026607 Clark Kimberling (ck6(AT)evansville.edu)
%I A052005
%S A052005 2,2,1,2,1,2,1,2,1,2,1,1,2,1,2,1,2,1,2,1,1,2,1,2,1,2,1,2,1,1,2,1,2,1,2,
%T A052005 1,1,2,1,2,1,2,1,2,1,1,2,1,2,1,2,1,2,1,1,2,1,2,1,2,1,1,2,1,2,1,2,1,2,1,
%U A052005 1,2,1,2,1,2,1,2,1,1,2,1,2,1,2,1,1,2,1,2,1,2,1,2,1,1,2,1,2,1,2,1,2,1,1
%N A052005 Number of Fibonacci numbers (A000045) with length n in base 2.
%C A052005 There are no double 2's except at the very start because multiplying by phi^3 adds at least 2 to Fn's binary length. For a similar reason there aren't any 3's because multiplying by phi^2 increments at least by one F(n)'s binary length.
%e A052005 F(17)= 1597{10} = 11000111101{2} the only one of length 11 and F(18)= 2584{10} = 101000011000{2} the only one of length 12 so both a(11) and a(12) equal 1.
%Y A052005 Cf. A052006, A000045, A050815, A036284, A037093, A022927, A022934.
%Y A052005 Adjacent sequences: A052002 A052003 A052004 this_sequence A052006 A052007 A052008
%Y A052005 Sequence in context: A137264 A078614 A026607 this_sequence A138702 A112104 A059426
%K A052005 nonn,base
%O A052005 1,1
%A A052005 Antti Karttunen and Patrick De Geest (pdg(AT)worldofnumbers.com), Nov 15 1999.
%I A138702
%S A138702 1,2,2,1,2,1,2,1,2,1,3,1,6,1,2,1
%N A138702 a(n) = number of terms in the continued fraction of the absolute value of B_n, the nth Bernoulli number.
%C A138702 The continued fraction terms being counted include the initial 0, if there is one. (a(n), for all odd n >= 3, is 1.)
%e A138702 The 12th Bernoulli number is -691/2730. Now 691/2730 has the continued fraction 0 + 1/(3 + 1/(1 + 1/(19 + 1/(3 + 1/11)))), which has 6 terms (including the zero). So a(12) = 6.
%Y A138702 Cf. A138701, A138703.
%Y A138702 Adjacent sequences: A138699 A138700 A138701 this_sequence A138703 A138704 A138705
%Y A138702 Sequence in context: A078614 A026607 A052005 this_sequence A112104 A059426 A082389
%K A138702 more,nonn
%O A138702 0,2
%A A138702 Leroy Quet (qq-quet(AT)mindspring.com), Mar 26 2008
%I A112104
%S A112104 1,2,2,1,2,1,2,2,1,2,1,2,2,2,2,2,1,1,1,1,1,2,2,1,1,1,2,2,1,2,1,2,2,1,1,
%T A112104 1,1,1,2,2,2,1,1,1,2,2,2,2,1,1,2,2,1,2,2,2,2,1,1,1,2,1,1,2,1,1,2,2,2,2,
%U A112104 1,1,1,2,2,2,2,2,1,1,2,2,1,2,2,2,2,1,2,2,2,2,1,2,1,1,1,2,1,2,1,2,2,2,1
%N A112104 Unique sequence of 1's and 2's where g.f. A(x) satisfies A(x) = B(B(x)) such that B(x) is an integer series, with A(0) = 0.
%H A112104 Paul D. Hanna, Table of n, a(n) for n = 1..512
%e A112104 G.f.: A(x) = x + 2*x^2 + 2*x^3 + x^4 + 2*x^5 + x^6 +...
%e A112104 then A(x) = B(B(x)) where
%e A112104 B(x) = x + x^2 + x^5 - 3*x^6 + 7*x^7 - 10*x^8 - 5*x^9 +...
%e A112104 is the g.f. of A112105.
%o A112104 (PARI) {a(n,m=2)=local(F=x+x^2+x*O(x^n),G);if(n<1,0, for(k=3,n, G=F+x*O(x^k);for(i=1,m-1,G=subst(F,x,G)); F=F-((polcoeff(G,k)-1)\m)*x^k); G=F+x*O(x^n);for(i=1,m-1,G=subst(F,x,G)); return(polcoeff(G,n,x)))}
%Y A112104 Cf. A112105, A112106-A112127.
%Y A112104 Adjacent sequences: A112101 A112102 A112103 this_sequence A112105 A112106 A112107
%Y A112104 Sequence in context: A026607 A052005 A138702 this_sequence A059426 A082389 A119469
%K A112104 nonn
%O A112104 1,2
%A A112104 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 27 2005
%I A059426
%S A059426 1,2,2,1,2,1,2,2,1,2,2,1,2,1,2,2,1,2,1,2,2,1,2,2,1,2,1,2,2,1,2,2,1,2,1,
%T A059426 2,2,1,2,1,2,2,1,2,2,1,2,1,2,2,1,2,1,2,2,1,2,2,1,2,1
%N A059426 First differences of A026273.
%C A059426 Differences of ranks of "2 between two 1": 5 8 5 8 8 5 8 5 8 8 5 8 8 5 8 5 8 8 5 8 5 8 8 5 8 8 5 8 5 8 8 5 8 8 5 8 5 8 8 5 8 5 8 8 5 8 8 5 8 5 8 8 5 8 5 8 8 5 8 8 5 8 5 8 8 5 8 8 5 8 5 8 8 5 8 5 8 8 5 8 8 5 8 5 8 8 5 8 8 5 8 5 8 8 5 8 5 8 8 5 8 8 5 8 5 8 8 5 8 5 8 8 5 8 8 5 8 5 8 8 5 8 8 5 8 5 8 8 5 8 5 8 8 5 8 ... Differences of ranks of "8 between two 5": 5 8 5 8 8 5 8 5 8 8 5 8 8 5 8 5 8 8 5...(self-generated?)
%Y A059426 Adjacent sequences: A059423 A059424 A059425 this_sequence A059427 A059428 A059429
%Y A059426 Sequence in context: A052005 A138702 A112104 this_sequence A082389 A119469 A127439
%K A059426 easy,nonn
%O A059426 0,2
%A A059426 Claude Lenormand (claude.lenormand(AT)free.fr), Jan 31 2001
%I A082389
%S A082389 1,2,2,1,2,1,2,2,1,2,2,1,2,1,2,2,1,2,1,2,2,1,2,2,1,2,1,2,2,1,2,2,1,2,1,
%T A082389 2,2,1,2,1,2,2,1,2,2,1,2,1,2,2,1,2,1,2,2,1,2,2,1,2,1,2,2,1,2,2,1,2,1,2,
%U A082389 2,1,2,1,2,2,1,2,2,1,2,1,2,2,1,2,2,1,2,1,2,2,1,2,1,2,2,1,2,2,1,2,1,2,2
%N A082389 a(n)=floor((n+2)*phi)-floor((n+1)*phi) where phi=(1+sqrt(5))/2.
%C A082389 Alternative descriptions (1): unique positive integer sequence taking values in {1,2} satisfying a(1)=1, a(2)=2 and a(a(1)+...+a(n))=a(n) for n >= 3.
%C A082389 (2) Start with 1,2; then for any k>=1, a(a(1)+...+a(k))=a(k), fill in any undefined terms by the rule that a(t) = 1 if a(t-1) = 2, and a(t) = 2 if a(t-1) = 1.
%C A082389 (3) a(1)= 1, a(2)=2, a(a(1)+a(2)+...+a(n))=a(n); a(a(1)+a(2)+...+a(n)+1)=3-a(n).
%C A082389 More generally, the sequence a(n)=floor(r*(n+2))-floor(r*(n+1)), r= (1/2) *(z+sqrt(z^2+4)), z integer >=1, is defined by a(1), a(2) and a(a(1)+a(2)+...+a(n)+f(z))=a(n); a(a(1)+a(2)+...+a(n)+f(z)+1)=(2z+1)-a(n) where f(1)=0, f(z)=z-2 for z>=2.
%F A082389 a(n) = A014675(n+1); sum(k = 1, n, a(k)) = A058065(n)
%F A082389 Apparently a(n) = A059426(n).
%e A082389 a(1)+a(2)=3 and a(a(1)+a(2)) must be a(2) so a(3)=2. Therefore a(a(1)+a(2)+a(3))=a(5)=2 and from the rule the "hole" a(4) is 1. Hence sequence begins 1,2,2,1,2,...
%t A082389 Rest@Nest[ Flatten[ # /. {1 -> 2, 2 -> {2, 1}}] &, {1}, 11] (from Robert G. Wilson v (rgwv(at)rgwv.com), Jan 26 2006)
%Y A082389 Same as A014675 without the first term.
%Y A082389 Adjacent sequences: A082386 A082387 A082388 this_sequence A082390 A082391 A082392
%Y A082389 Sequence in context: A138702 A112104 A059426 this_sequence A119469 A127439 A092788
%K A082389 nonn,nice,easy
%O A082389 1,2
%A A082389 Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 14 2003
%I A119469
%S A119469 1,1,2,2,1,2,1,2,2,1,2,2,1,2,1,2,2,3,1,2,2,1,2,2,3,3,3,2,2,1,2,1,2,2,3,
%T A119469 3,5,3,4,3,3,1,2,2,1,2,2,3,3,5,5,5,5,5,4,5,3,2,2,1,2,1,2,2,3,3,5,5,
%U A119469 7,6,7,6,8,6,7,5,5,3,3,1,2,2,1,2,2,3,3,5,5,7,8,8,8,10,9,10,10,10,8
%N A119469 Triangle read by rows: row n gives coefficients (lowest degree first) of P_n(x), where P_0(x) = P_1(x) = 1; P_n(x) = P_{n-1}(x) + x^(n-2)*P_{n-2}(x).
%C A119469 P_n(x) has degree A002620(n).
%e A119469 Triangle begins:
%e A119469 1
%e A119469 1
%e A119469 2
%e A119469 2,1
%e A119469 2,1,2
%e A119469 2,1,2,2,1
%e A119469 2,1,2,2,3,1,2
%e A119469 2,1,2,2,3,3,3,2,2,1
%e A119469 2,1,2,2,3,3,5,3,4,3,3,1,2
%e A119469 2,1,2,2,3,3,5,5,5,5,5,4,5,3,2,2,1
%p A119469 P[0]:=1; P[1]:=1; d:=[0,0]; M:=14; for n from 2 to M do P[n]:=expand(P[n-1]+q^(n-2)*P[n-2]);
%p A119469 lprint(seriestolist(series(P[n],q,M^2))); d:=[op(d),degree(P[n],q)]; od: d;
%Y A119469 A variant of A127836.
%Y A119469 Rows converge to A003113.
%Y A119469 Adjacent sequences: A119466 A119467 A119468 this_sequence A119470 A119471 A119472
%Y A119469 Sequence in context: A112104 A059426 A082389 this_sequence A127439 A092788 A091267
%K A119469 nonn,tabf
%O A119469 0,3
%A A119469 njas, Apr 10 2007
%I A127439
%S A127439 2,2,1,2,1,2,2,1,2,2,2,1,2,2,4,2,1,2,2,4,2,2,1,2,2,4,2,4,2,1,2,2,4,2,4,
%T A127439 2,2,1,2,2,4,2,4,2,4
%N A127439 Triangle read by rows, in which row n consists of first n terms of A054541.
%C A127439 Row sums = p(n) (A000040).
%C A127439 Rows tend to A054541: (2, 1, 2, 2, 4, 2, 4,...)
%e A127439 p(5) = 11 = (2 + 1 + 2 + 2 + 4).
%Y A127439 Cf. A054541, A000040, A001223.
%Y A127439 Adjacent sequences: A127436 A127437 A127438 this_sequence A127440 A127441 A127442
%Y A127439 Sequence in context: A059426 A082389 A119469 this_sequence A092788 A091267 A003643
%K A127439 nonn,tabl
%O A127439 1,1
%A A127439 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 14 2007
%E A127439 Edited by njas, Jan 27 2007
%I A092788
%S A092788 1,1,1,1,2,2,1,2,1,2,2,2
%N A092788 USUP perfect numbers.
%C A092788 USUP stands for UnitarySigmaUnitaryPhi(n) or USUP(n).
%C A092788 Consider numbers such that USUP(n) = n/k. If n = Product p_i^r_i then USUP(n) = UnitarySigma(2^r_1)*UnitaryPhi(n/2^r_1) = (2^(r_1)+1)*Product(p_i^r_i-1), 2Index entries for sequences related to quadratic fields
%Y A003643 Adjacent sequences: A003640 A003641 A003642 this_sequence A003644 A003645 A003646
%Y A003643 Sequence in context: A127439 A092788 A091267 this_sequence A058062 A102820 A024317
%K A003643 nonn,nice
%O A003643 1,4
%A A003643 njas, Mira Bernstein
%I A058062
%S A058062 0,1,1,1,2,2,1,2,1,2,2,2,2,2,2,1,2,2,2,3,1,2,2,3,1,3,2,2,3,2,1,2,2,2,2,
%T A058062 2,2,3,2,3,3,2,2,3,3,2,2,2,2,2,2,2,2,3,2,3,2,3,3,3,2,2,2,1,3,2,2,3,2,2,
%U A058062 2,3,2,3,2,3,2,3,2,3,1,3,3,2,2,3,3,3,3,3,2,3,1,2,3,3,2,2,3,2,3,2,2,4,2
%N A058062 Number of distinct prime factors of sigma(n), the sum of the divisors of n.
%F A058062 a(n)=A001221(A000203(n))
%e A058062 n=81, sigma(81)=81+27+9+3+1=121, so a(81)=1
%Y A058062 Cf. A001221, A000203.
%Y A058062 Adjacent sequences: A058059 A058060 A058061 this_sequence A058063 A058064 A058065
%Y A058062 Sequence in context: A092788 A091267 A003643 this_sequence A102820 A024317 A024880
%K A058062 nonn
%O A058062 0,5
%A A058062 Labos E. (labos(AT)ana.sote.hu), Nov 23 2000
%I A102820
%S A102820 1,1,2,2,1,2,1,2,2,2,3,1,1,1,3,3,0,2,2,0,3,1,2,4,2,0,1,0,1,6,1,3,1,3,0,
%T A102820 3,1,1,1,3,1,3,1,1,1,4,4,1,1,2,1,1,3,2,2,0,1,1,1,1,3,6,2,0,1,6,1,3,0,1,
%U A102820 1,3,2,2,1,2,1,1,2,4,1,3,1,1,2,1,2,1,0,1,4,2,1,3,0,2,5,0,5,3,3,2,1,0,2
%N A102820 Number of primes between 2*p(n) and 2*p(n+1), where p(n) is the n-th prime.
%e A102820 a(15)=3 because there are 3 primes between the doubles of the 15th and 16th primes, that is between 2*47 and 2* 53.
%t A102820 MyA102820=Table[PrimePi[2 Prime[n+1]]-PrimePi[2 Prime[n]], {n, 150}] (Seidov)
%t A102820 f[n_] := PrimePi[2Prime[n + 1]] - PrimePi[2Prime[n]]; Table[ f[n], {n, 105}] (from Robert G. Wilson v Mar 03 2005)
%Y A102820 Cf. A104380.
%Y A102820 Adjacent sequences: A102817 A102818 A102819 this_sequence A102821 A102822 A102823
%Y A102820 Sequence in context: A091267 A003643 A058062 this_sequence A024317 A024880 A029424
%K A102820 easy,nonn
%O A102820 1,3
%A A102820 Ali A. Tanara (tanara(AT)khayam.ut.ac.ir), Feb 27 2005
%E A102820 More terms from Zak Seidov (zakseidov(AT)yahoo.com), Feb 28 2005
%I A024317
%S A024317 0,0,0,1,1,0,1,1,1,1,2,2,1,2,1,2,2,2,3,3,2,2,3,2,3,3,3,2,4,4,3,4,3,4,3,3,4,
%T A024317 4,3,4,5,5,4,5,4,4,5,3,5,5,5,4,5,5,5,6,5,5,6,6,5,5,5,6,6,5,5,6,5,6,7,7,5,7,
%U A024317 7,7,7,4,7,6,6,7,7,6,6,7,7,7,8,7
%N A024317 s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n+1-k), where k=[ (n+1)/2) ], s = A023531, t = A023532.
%Y A024317 Adjacent sequences: A024314 A024315 A024316 this_sequence A024318 A024319 A024320
%Y A024317 Sequence in context: A003643 A058062 A102820 this_sequence A024880 A029424 A061498
%K A024317 nonn
%O A024317 1,11
%A A024317 Clark Kimberling (ck6(AT)evansville.edu)
%I A024880
%S A024880 0,0,1,1,0,1,1,1,1,2,2,1,2,1,2,2,2,3,3,2,2,3,2,3,3,3,2,4,4,3,4,3,4,3,3,4,4,
%T A024880 3,4,5,5,4,5,4,4,5,3,5,5,5,4,5,5,5,6,5,5,6,6,5,5,5,6,6,5,5,6,5,6,7,7,5,7,7,
%U A024880 7,7,4,7,6,6,7,7,6,6,7,7,7,8,7,6,8,8,7,8,7,7,7,7,8,8,8,6,8,7,8,8,7,8,9,8,8
%N A024880 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = A023531, t = A023532.
%Y A024880 Adjacent sequences: A024877 A024878 A024879 this_sequence A024881 A024882 A024883
%Y A024880 Sequence in context: A058062 A102820 A024317 this_sequence A029424 A061498 A106029
%K A024880 nonn
%O A024880 2,10
%A A024880 Clark Kimberling (ck6(AT)evansville.edu)
%I A029424
%S A029424 1,0,0,0,0,0,1,0,1,1,0,1,1,0,1,1,1,2,2,1,2,1,2,2,3,2,3,
%T A029424 3,3,3,4,3,4,5,4,5,6,4,6,6,6,7,8,6,9,8,8,9,10,9,11,11,11,
%U A029424 12,13,12,14,14,14,15,17,15,18,18,18,19,21,19,22,22,22
%N A029424 Expansion of 1/((1-x^6)(1-x^8)(1-x^9)(1-x^11)).
%Y A029424 Adjacent sequences: A029421 A029422 A029423 this_sequence A029425 A029426 A029427
%Y A029424 Sequence in context: A102820 A024317 A024880 this_sequence A061498 A106029 A105153
%K A029424 nonn
%O A029424 0,18
%A A029424 njas
%I A061498
%S A061498 0,0,1,1,1,1,1,1,2,2,1,2,1,2,3,1,1,2,1,2,3,2,1,2,2,2,2,2,1,3,1,1,3,2,3,
%T A061498 2,1,2,3,2,1,3,1,2,3,2,1,2,2,2,3,2,1,2,3,2,3,2,1,3,1,2,3,1,3,3,1,2,3,3,
%U A061498 1,2,1,2,3,2,3,3,1,2,2,2,1,3,3,2,3,2,1,3,3,2,3,2,3,2,1,2,3,2,1,3,1,2,5
%N A061498 Number of distinct terms in the first difference sequence of reduced residue system[=dRRS] for n.
%e A061498 For n=24, RRS={1, 5, 7, 11, 13, 17, 19, 23}, dRRS={4, 2, 4, 2, 4, 2, 4}; for n=32, dRRS={2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}; for n=31, dRRS={1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}; for n=34, RRS={1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25, 27, 29, 31, 33}, dRRS={2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2} so a(24)=2, a(32)=1, a(31)=1, a(34)=2.
%Y A061498 Adjacent sequences: A061495 A061496 A061497 this_sequence A061499 A061500 A061501
%Y A061498 Sequence in context: A024317 A024880 A029424 this_sequence A106029 A105153 A000924
%K A061498 nonn
%O A061498 0,9
%A A061498 Labos E. (labos(AT)ana.sote.hu), Jun 12 2001
%I A106029
%S A106029 1,1,1,2,2,1,2,1,2,3,2,3,1,4,2,2,4,4,4,2,4,3,2,2,4,3,5,4,1,3,3,2,4,3,4,
%T A106029 2,2,4,5,6,6,1,6,4,4,3,6,6,4,3,3,2,4,6,4,7,2,4,5,5,3
%N A106029 a(n) is the number of orbits under the action of GL_2[Z] on the primitive binary quadratic forms of discriminant D, where D=m if m=1 (mod 4), D=4*m otherwise and m<0 is the n-th squarefree number.
%C A106029 A000924 is the same except it is under the action of SL_2[Z].
%H A106029 S. R. Finch, Class number theory
%H A106029 Jens Jonasson, Classes of integral binary quadratic forms, Masters thesis (2001), Appendix B.
%e A106029 m=-1, -2, -3, -5, -6, -7, -10, -11, -13, ...
%e A106029 with corresponding discriminant
%e A106029 D=-4, -8, -3, -20, -24, -7, -40, -11, -52, ....
%Y A106029 Cf. A000924.
%Y A106029 Adjacent sequences: A106026 A106027 A106028 this_sequence A106030 A106031 A106032
%Y A106029 Sequence in context: A024880 A029424 A061498 this_sequence A105153 A000924 A109909
%K A106029 nonn
%O A106029 1,4
%A A106029 S. R. Finch (Steven.Finch(AT)inria.fr), May 05 2005
%I A105153
%S A105153 1,1,2,2,1,2,1,2,4,2,2,4,4,2,2,4,4,1,4,4,4,1,4,4,4,1,4,4,4,1,4,4,4,4,8,
%T A105153 4,4,4,4,8,4,4,8,4,4,4,4,8,4,4,8,4,4,4,4,8,4,4,8,4,4,4,4,8,2,16,8,4,2,4,
%U A105153 8,16,2,16,8,4,2,4,8,16,2,16,8,4,2,4,8,16,2,16,8,4,2,4,8,16,2,16,8,4,2
%N A105153 Consider trajectory of n under repeated application of map k -> A105025(k); a(n) = length of cycle.
%C A105153 Why is this always a power of 2?
%H A105153 David Applegate, Benoit Cloitre, Philippe DELEHAM and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
%o A105153 (C++) #include #include #include using namespace std ; int main(int argc, char *argv[]) { int kmax = 8 ; vector a105025; a105025.push_back(0) ; a105025.push_back(1) ; for(int k=1 ; k < kmax ; k++) { int bstrt = 1 << k ; for(int j=0 ; j < bstrt ; j++) { int s = bstrt ; for(int i= k-1 ; i >=0 ; i--) s += (bstrt+j+k-i) & ( 1< traj ; while ( traj.find(nrep) == traj.end() ) { traj.insert(nrep) ; if ( nrep < a105025.size() ) nrep = a105025[nrep] ; else break ; } cout << traj.size() << "," ; } cout << endl ; return 0 ; } - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 10 2007
%Y A105153 Positions of 1's: A105271.
%Y A105153 Cf. A102370, A105025, A105027, A105154.
%Y A105153 Adjacent sequences: A105150 A105151 A105152 this_sequence A105154 A105155 A105156
%Y A105153 Sequence in context: A029424 A061498 A106029 this_sequence A000924 A109909 A030768
%K A105153 nonn,easy
%O A105153 0,3
%A A105153 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 30 2005
%E A105153 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 10 2007
%I A000924 M0195 N0072
%S A000924 1,1,1,2,2,1,2,1,2,4,2,4,1,4,2,3,6,6,4,3,4,4,2,2,6,4,8,4,1,4,5,2,6,4,4,
%T A000924 2,3,6,8,8,8,1,8,4,7,4,10,8,4,5,4,3,4,10,6,12,2,4,8,8,4,14,4,5,8,6,3,6,
%U A000924 12,8,8,8,2,6,10,10,2,5,12,4,5,4,14,8,8,3,8,4,10,8,16,14,7,8,4,6,8,10
%N A000924 Class number of Q(sqrt(-n)), n square-free.
%D A000924 Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 425-430.
%D A000924 D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
%D A000924 R. A. Mollin, Quadratics, CRC Press, 1996, Appendix D, gives a table for n <= 1999, correcting that of Borevich and Shafarevich.
%H A000924 T. D. Noe, Table of n, a(n) for n=1..10000
%H A000924 S. R. Finch, Class number theory
%H A000924 Index entries for sequences related to quadratic fields
%e A000924 a(10)=4, since 14 is the 10-th squarefree number and the class number of Q(sqrt(-14)) is 4.
%Y A000924 Values of n run through A005117. Corresponding discriminants give A033197.
%Y A000924 Adjacent sequences: A000921 A000922 A000923 this_sequence A000925 A000926 A000927
%Y A000924 Sequence in context: A061498 A106029 A105153 this_sequence A109909 A030768 A051480
%K A000924 nonn,nice,easy
%O A000924 1,4
%A A000924 njas, Mira Bernstein
%E A000924 Edited by Dean Hickerson (dean(AT)math.ucdavis.edu), Mar 17 2003
%I A109909
%S A109909 0,0,0,2,2,1,2,1,4,1,3,2,3,2,3,2,4,3,5,1,10,1,5,5,4,2,6,3,5,3,9,4,11,3,
%T A109909 5,5,5,5,14,1,6,6,7,6,11,5,8,4,15,3,13,4,10,9,6,5,11,4,12,5,13,4,12,4,6,
%U A109909 11,13,4,12,6,15,12,9,4,9,5,10,8,10,3,28,7,11,15,6,9,20,7,20,6,17,5,23
%N A109909 a(n) = number of primes of the form k*(n-k)-1.
%C A109909 Conjecture: a(n) > 0 for n > 3.
%Y A109909 Cf. A109904, A109905, A109906, A109907, A109908.
%Y A109909 Adjacent sequences: A109906 A109907 A109908 this_sequence A109910 A109911 A109912
%Y A109909 Sequence in context: A106029 A105153 A000924 this_sequence A030768 A051480 A071572
%K A109909 nonn
%O A109909 1,4
%A A109909 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 15 2005
%E A109909 More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Aug 15 2005
%I A030768
%S A030768 2,2,1,2,1,4,3,2,1,5,4,3,2,1,7,6,5,4,3,2,1,9,8,7,6,5,4,3,2,1,
%T A030768 12,10,9,8,7,6,5,4,3,2,1,15,13,12,10,9,8,7,6,5,4,3,2,1,18,16,
%U A030768 15,13,12,10,9,8,7,6,5,4,3,2,1,21,19,18,16,15,13,12,11,10,9,8
%N A030768 Row 2, where, at stage k>1, write i in row 1 and j in row 2, where i is the number of j's in rows 1 and 2, for j=m,m-1,...2,1, where m=max number in row 1 from stages 1 to k-1; stage 1 is 2 in row 1.
%Y A030768 Adjacent sequences: A030765 A030766 A030767 this_sequence A030769 A030770 A030771
%Y A030768 Sequence in context: A105153 A000924 A109909 this_sequence A051480 A071572 A060256
%K A030768 nonn
%O A030768 1,1
%A A030768 Clark Kimberling (ck6(AT)evansville.edu)
%I A051480
%S A051480 2,2,1,2,1,4,6,1,2,1,1,1,1,1,1,3,1,3,1,4,1,1,2,1,1,3,2,4,1,3,2,1,1,2,2,
%T A051480 3,1,2,2,3,2,2,2,3,1,1,1,3,1,1,3,7,2,1,3,5,3,2,2,1,1,3,1,2,1,1,1,2,1,6,
%U A051480 1,2,1,2,3,5,1,3,1,3,1,1,1,2,2,2,5,2,2,3,3,1,1,8,1,5,1,1,3,1,5,1,1,1,1
%N A051480 Run lengths in binary expansion of Pi.
%e A051480 The binary expansion of Pi is: 11. 00 1 00 1 0000 111111 0 ...
%Y A051480 A004601.
%Y A051480 Adjacent sequences: A051477 A051478 A051479 this_sequence A051481 A051482 A051483
%Y A051480 Sequence in context: A000924 A109909 A030768 this_sequence A071572 A060256 A125916
%K A051480 base,nonn
%O A051480 1,1
%A A051480 H. Tracy Hall (hthall(AT)math.berkeley.edu)
%I A071572
%S A071572 1,2,2,1,2,1,5,1,1,2,1,1,2,1,1,1,2,1,1,3,1,1,1,1,1,2,1,2,1,1,1,1,3,1,2,
%T A071572 1,1,1,1,1,2,1,2,1,4,1,1,1,5,1,1,2,1,1,1,1,5,1,1,2,1,1,1,2,1,1,1,1,3,1,
%U A071572 1,1,1,1,1,1,1,1,1,1,2,1,2,1,1,1,1,1,3,1,1,1,1,1,1,1,1,3,1,1,1,1,1,5,1
%N A071572 Least k such that nextprime(k*prime(n)) - k*prime(n) is nonprime.
%C A071572 The first time a value of m appears is when n equals 1, 2, 20, 45, 7. I can find no value greater than 5 for n's up through 12,500,000.
%t A071572 NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; f[n_] := Block[{k = 1}, While[ PrimeQ[ NextPrim[k*Prime[n]] - k*Prime[n]], k++ ]; k]; Table[ f[n], {n, 1, 105}]
%Y A071572 Adjacent sequences: A071569 A071570 A071571 this_sequence A071573 A071574 A071575
%Y A071572 Sequence in context: A109909 A030768 A051480 this_sequence A060256 A125916 A100244
%K A071572 nonn
%O A071572 1,2
%A A071572 David W. Wilson (davidwwilson(AT)comcast.net) and Robert G. Wilson v (rgwv(AT)rgwv.com), May 31 2002
%I A060256
%S A060256 2,2,1,2,1,6,8,11,4,16,22,4,74,24,37,28,14,11,242,11,91,20,83,91,35,80,
%T A060256 48,47,226,2,12,203,30,38,356,54,266,429,305,227,173,1185,738,13,382,
%U A060256 277,455,433,173,1303
%N A060256 Smallest multiple a(n) of n-th primorial q(n) such that a(n)*q(n)-1 and a(n)*q(n)+1 are a pair of twin primes.
%e A060256 30030*j-1 or 30030*j+1 are not both primes for j=1,2,3,4,5. But for j=6 {180179,180181} are twin primes. So a(6)=6.
%Y A060256 Cf. A001359, A002110, A060229-A060232, A060255, A057706.
%Y A060256 Adjacent sequences: A060253 A060254 A060255 this_sequence A060257 A060258 A060259
%Y A060256 Sequence in context: A030768 A051480 A071572 this_sequence A125916 A100244 A059689
%K A060256 nonn
%O A060256 0,1
%A A060256 Labos E. (labos(AT)ana.sote.hu), Mar 22 2001
%I A125916
%S A125916 1,1,0,1,1,2,2,1,2,2
%N A125916 Sprague-Grundy values for octal game .15.
%C A125916 The sequence is eventually periodic with period 10. The only exception is at n=0 (not shown).
%C A125916 This game is called "Guiles" in Winning Ways.
%D A125916 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982; see Chapter 4, p. 104.
%Y A125916 Adjacent sequences: A125913 A125914 A125915 this_sequence A125917 A125918 A125919
%Y A125916 Sequence in context: A051480 A071572 A060256 this_sequence A100244 A059689 A131837
%K A125916 nonn
%O A125916 1,6
%A A125916 Richard Sabey (richardsabey(AT)hotmail.co.uk), Jan 24 2007
%I A100244
%S A100244 1,2,2,1,2,2,1,1,1,1,1,1,1,1,1,1,2,2,1,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T A100244 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A100244 1,1,1,1,1,1,1,1,1
%N A100244 a(n) = smallest positive integer such that {1 + product{k=1 to n} a(k)} is coprime to n.
%C A100244 a(22) to a(256) are each equal to 1. a(257) = 2.
%Y A100244 Adjacent sequences: A100241 A100242 A100243 this_sequence A100245 A100246 A100247
%Y A100244 Sequence in context: A071572 A060256 A125916 this_sequence A059689 A131837 A087889
%K A100244 nonn
%O A100244 1,2
%A A100244 Leroy Quet (qq-quet(AT)mindspring.com), Jan 11 2005
%I A059689
%S A059689 1,1,1,1,2,2,1,2,2,1,1,1,1,1,2,1,1,3,1,2,1,1,2,2,1,2,3,1,3,1,1,3,2,1,3,
%T A059689 3,2,1,1,2,1,2,2,1,3,2,2,1,2,2,2,2,2,3,2,3,1,2,3,2,2,3,3,3,1,1
%N A059689 List consisting of all functions from [1,2,3,...,n] into [1,2,3,...,n] for n=1,2,3,... (written out in full) in lexicographic order.
%C A059689 Includes every finite sequence of positive integers. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jan 26 2006
%e A059689 For n=1 there is one function, 1. For n=2 there are 4 such functions and ordering these lexicographically we get (1,1), (1,2), (2,1), (2,2). Thus the sequence begins 1; (1,1), (1,2), (2,1), (2,2); ...
%Y A059689 A030298.
%Y A059689 Adjacent sequences: A059686 A059687 A059688 this_sequence A059690 A059691 A059692
%Y A059689 Sequence in context: A060256 A125916 A100244 this_sequence A131837 A087889 A014710
%K A059689 nonn
%O A059689 1,5
%A A059689 Avi Peretz (njk(AT)netvision.net.il), Feb 06 2001
%I A131837
%S A131837 0,0,0,2,2,1,2,2,1,1,1,2,2,1,3,2,1,2,2,2,1,1,3,2,1,1,1,2,2,2,2,1,2,2,2,
%T A131837 1,1,1,1,1,1,1,2,2,1,2,2,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,
%U A131837 1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A131837 Multiplicative persistence of Cullen numbers.
%C A131837 After the 111st element all the numbers have some digits equal to zero thus the persistence is equal to 1.
%e A131837 Cullen number 65 --> 6*5=30 --> 3*0=0 thus persistence is 2.
%p A131837 P:=proc(n) local i,k,w,ok,cont; for i from 0 by 1 to n do w:=1; k:=i*2^i+1; ok:=1; if k<10 then print(0); else cont:=1; while ok=1 do while k>0 do w:=w*(k-(trunc(k/10)*10)); k:=trunc(k/10); od; if w<10 then ok:=0; print(cont); else cont:=cont+1; k:=w; w:=1; fi; od; fi; od; end: P(120);
%Y A131837 Cf. A002064, A131840.
%Y A131837 Adjacent sequences: A131834 A131835 A131836 this_sequence A131838 A131839 A131840
%Y A131837 Sequence in context: A125916 A100244 A059689 this_sequence A087889 A014710 A055174
%K A131837 easy,nonn,base
%O A131837 1,4
%A A131837 Paolo P. Lava & Giorgio Balzarotti (ppl(AT)spl.at), Jul 20 2007
%I A087889
%S A087889 2,2,1,2,2,1,1,2,1,2,2,1,2,1,1,2,2,1,2,2,1,1,2,1,1,2,1,2,2,1,2,1,1,2,2,
%T A087889 1,2,2,1,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2,1
%N A087889 Given a sequence u consisting just of 1's and 2's, let f(u)(n) be the length of n-th run. Then we may define a sequence u = {a(n)} by a(n)=f^(n-1)(u)(1) (starting with n=1).
%C A087889 There are exactly three infinite sequences satisfying this relation, namely this sequence, A087888 and A087890.
%Y A087889 Cf. A000002, A087888, A087890.
%Y A087889 Adjacent sequences: A087886 A087887 A087888 this_sequence A087890 A087891 A087892
%Y A087889 Sequence in context: A100244 A059689 A131837 this_sequence A014710 A055174 A096369
%K A087889 easy,eigen,nonn
%O A087889 1,1
%A A087889 Vincent Nesme (vincent.nesme(AT)ens-lyon.fr), Oct 13 2003
%E A087889 The description was not quite clear to me but I hope I have edited it correctly. - njas
%I A014710
%S A014710 2,2,1,2,2,1,1,2,2,2,1,1,2,1,1,2,2,2,1,2,2,1,1,1,2,2,1,1,2,1,1
%N A014710 The regular paper-folding (or dragon curve) sequence.
%D A014710 G. Melancon, Factorizing infinite words using Maple, MapleTech journal, vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.
%H A014710 Index entries for sequences obtained by enumerating foldings
%F A014710 Set a=2, b=1, S(0)=a, S(n+1) = S(n)aF(S(n)), where F(x) reverses x and then interchanges a and b; sequence is limit S(infinity).
%F A014710 a(4n) = 2, a(4n+2) = 1, a(2n+1) = a(n).
%Y A014710 See A014577 for more references and more terms.
%Y A014710 Adjacent sequences: A014707 A014708 A014709 this_sequence A014711 A014712 A014713
%Y A014710 Sequence in context: A059689 A131837 A087889 this_sequence A055174 A096369 A102297
%K A014710 nonn
%O A014710 0,1
%A A014710 njas
%I A055174
%S A055174 2,2,1,2,2,1,1,2,4,1,3,2,5,1,4,4,1,3,1,2,6,1,7,4,3,3,2,5,1,2,8,1,9,4,4,
%T A055174 3,4,5,2,6,1,7,1,2,10,1,12,4,7,3,5,5,3,6,2,7,2,8,1,9,1,2,13,1,15,4,8,3,
%U A055174 7,5,5,6,3,7,4,8,2,9,2,10,1,12,1,2,16,1,18
%N A055174 Cumulative counting sequence: method B (noun,adjective)-pairs with first term 2.
%C A055174 Segments (as in %e): 2; 2,1; 2,2,1,1; 2,4,1,3,2,5; ...
%C A055174 Conjecture: every positive integer occurs.
%e A055174 Write 2, thus having 2 once, thus having 2 twice and 1 once, thus having 2 4 times and 1 3 times and 2 5 times, etc.
%Y A055174 Adjacent sequences: A055171 A055172 A055173 this_sequence A055175 A055176 A055177
%Y A055174 Sequence in context: A131837 A087889 A014710 this_sequence A096369 A102297 A098398
%K A055174 nonn
%O A055174 1,1
%A A055174 Clark Kimberling (ck6(AT)evansville.edu), Apr 27 2000
%I A096369
%S A096369 0,1,2,2,1,2,2,1,1,2,5,3,3,2,5,7,3,4,5,3,7,13,7,6,6,4,7,13,23,13,12,9,
%T A096369 10,12,11,23,43,22,23,22,23,22,21,21,43,75,37,37,36,40,39,38,38,37,75,
%U A096369 137,71,71,73,66,56,71,70,66,67,137,255,128,125,130,127,132,128,130,129
%N A096369 Triangle read by rows, 0<=k1; T(n,n-1) = T(n,0) for n>2;
%C A096369 T(n,1) = A095008(n-1) for n>2;
%C A096369 T(n,n-2) = A095766(n-1) for n>1;
%C A096369 conjecture: T(n,k) > 0 for n>1.
%e A096369 prime(12) = 37 -> 1 0 0 1 0 1 ..... n = 6
%e A096369 prime(13) = 41 -> 1 0 1 0 0 1 ..... all primes p:
%e A096369 prime(14) = 43 -> 1 0 1 0 1 1 ..... 2^(6-1) <= p < 2^6
%e A096369 prime(15) = 47 -> 1 0 1 1 1 1
%e A096369 prime(16) = 53 -> 1 1 0 1 0 1
%e A096369 prime(17) = 59 -> 1 1 1 0 1 1
%e A096369 prime(18) = 61 -> 1 1 1 1 0 1
%e A096369 col-sums of bits: 7 3 5 4 3 7 : T(6,5)=7, T(6,4)=3, T(6,3)=5,
%e A096369 ...
%Y A096369 Adjacent sequences: A096366 A096367 A096368 this_sequence A096370 A096371 A096372
%Y A096369 Sequence in context: A087889 A014710 A055174 this_sequence A102297 A098398 A131714
%K A096369 nonn,tabl
%O A096369 1,3
%A A096369 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 19 2004
%I A102297
%S A102297 1,2,2,1,2,2,1,2,1,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,1,2,3,2,3,2,2,2,
%T A102297 3,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,2,2,2,2,2,2,2,2,3,1,2,2,2,1,3,2,3,2,2,
%U A102297 2,2,2,2,2,2,2,2,2,2,2,3,2,3,2,2,2,2,3,2,1,3,2,2,2,2,2,2,2,2,2,3,2,2,2
%N A102297 Number of distinct divisors of n+1 where n and n+1 are composite or twin composite numbers.
%C A102297 It is interesting to note that the first such consecutive pair of composite numbers is 8 and 9 which are perfect powers: 2^3 and 3^2. Conjecture: 8 and 9 are the only 2 consecutive composite numbers that are both perfect powers. Or, if x>2, x^m+1 != y^n for all m,n,x,y. Now if we relax the condition that 0 and 1 are not composite, we have 0^m+1 = 1^n for all m,n an infinity of solutions.
%e A102297 For n=8 n+1 = 9 = 3*3 or 1 distinct divisor.
%o A102297 (PARI) f(n) = for(x=1,n,y=composite(x)+1;if(!isprime(y),print1(omega(y)","))) composite(n) =\The n-th composite number. 1 is def as not prime nor composite. { local(c,x); c=1; x=1; while(c <= n, x++; if(!isprime(x),c++); ); return(x) }
%Y A102297 Adjacent sequences: A102294 A102295 A102296 this_sequence A102298 A102299 A102300
%Y A102297 Sequence in context: A014710 A055174 A096369 this_sequence A098398 A131714 A130196
%K A102297 easy,nonn
%O A102297 1,2
%A A102297 Cino Hilliard (hillcino368(AT)gmail.com), Feb 19 2005
%I A098398
%S A098398 1,1,1,1,1,1,2,2,1,2,2,1,2,2,1,1,2,2,1,2,2,1,1,1,1,2,2,2,2,1,1,1,2,2,2,
%T A098398 2,1,1,1,1,2,2,2,2,2,2,1,1,2,2,1,2,2,1,1,1,2,2,1,2,2,1,1,2,2,1,1,1,2,2,
%U A098398 1,1,1,1,1,1,1,1,1,1,2,2,2,2,1,1,1,1,2,2,1,1,1,1,1,1,1,2,2,1,1,1,1,2,2
%N A098398 Number of primes that are not less than prime(n)-Log2(Log2(prime(n))) and not greater than prime(n)+Log2(Log2(prime(n))), where Log2=A000523.
%C A098398 a(n) = A000720(A098392(n)) - A000720(A098393(n)-1);
%C A098398 a(n) <= A098396(n) <= A098397(n) <= A097935(n);
%C A098398 a(n)<=2 for n<=6543; a(6544)=#{2^16+1=65537,65539,65543}=3.
%e A098398 a(10) = #{p prime: A098392(10) <= p <= A098393(10)} =
%e A098398 = #{p prime: 27 <= p <= 31} = #{29,31} = 2.
%Y A098398 Adjacent sequences: A098395 A098396 A098397 this_sequence A098399 A098400 A098401
%Y A098398 Sequence in context: A055174 A096369 A102297 this_sequence A131714 A130196 A119646
%K A098398 nonn
%O A098398 1,7
%A A098398 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Sep 06 2004
%I A131714
%S A131714 1,2,2,1,2,2,1,2,2,1,2,2,1,2,2,1,2,2,1,2,2,1,2,2
%V A131714 1,-2,2,-1,2,-2,1,-2,2,-1,2,-2,1,-2,2,-1,2,-2,1,-2,2,-1,2,-2
%N A131714 Period 6: repeat 1,-2,2,-1,2,-2.
%F A131714 a(n)=(1/6)*{-3*(n mod 6)+4*[(n+1) mod 6]-3*[(n+2) mod 6]+3*[(n+3) mod 6]-4*[(n+4) mod 6]+3*[(n+5) mod 6]}, with n>=0 - Paolo P. Lava (ppl(AT)spl.at), Oct 02 2007
%F A131714 G.f.: (1-2*x+2*x^2)/(x+1)/(x^2-x+1). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 14 2007
%Y A131714 Cf. A130196, A131556.
%Y A131714 Adjacent sequences: A131711 A131712 A131713 this_sequence A131715 A131716 A131717
%Y A131714 Sequence in context: A096369 A102297 A098398 this_sequence A130196 A119646 A024693
%K A131714 sign
%O A131714 0,2
%A A131714 Paul Curtz (bpcrtz(AT)free.fr), Sep 14 2007
%I A130196
%S A130196 1,2,2,1,2,2,1,2,2,1,2,2,1,2,2,1,2,2,1,2,2,1,2,2,1,2,2,1,2,2,1,2,2,1,2,
%T A130196 2,1,2,2,1,2,2,1,2,2,1,2,2,1,2,2,1,2,2,1,2,2,1,2,2,1,2,2,1,2,2,1,2,2,1,
%U A130196 2,2,1,2,2,1,2,2,1,2,2,1,2,2
%N A130196 Period 3: repeat 1 2 2.
%F A130196 a(0)=1,a(1)=a(2)=2,a(n+3)=a(n).
%F A130196 a(n)=(1/9)*{8*(n mod 3)+5*[(n+1) mod 3]+2*[(n+2) mod 3]}, with n>=0. - Paolo P. Lava (ppl(AT)spl.at), Aug 28 2007
%F A130196 G.f.: -(1+2*x+2*x^2)/(x-1)/(x^2+x+1). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 14 2007
%Y A130196 Adjacent sequences: A130193 A130194 A130195 this_sequence A130197 A130198 A130199
%Y A130196 Sequence in context: A102297 A098398 A131714 this_sequence A119646 A024693 A025126
%K A130196 nonn
%O A130196 0,2
%A A130196 Paul Curtz (bpcrtz(AT)free.fr), Aug 05 2007
%I A119646
%S A119646 1,1,1,1,1,1,1,1,1,2,2,1,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%T A119646 3,2,2,2,2,2,3,2,2,2,3,2,2,2,3,2,2,2,2,2,2,2,2,3,2,3,3,2,2,2,2,3,3,2,2,
%U A119646 3,3,3,3,3,3,2,3,3,3,3,3,2,3,3,3,3,3,3,3,3,3,2,3,3,3,3,3,2,3,3,3,3,3,3
%N A119646 Number of partitions into 3 parts with the same product.
%e A119646 See A119028.
%t A119646 pdt[lst_] := lst[[1]]*lst[[2]]*lst[[3]];
%t A119646 tanya[n_] := Max[Length /@ Split[Sort[pdt /@ Union[ Partition[Last /@ Flatten[ FindInstance[a + b + c == n && a >= b >= c > 0, {a, b, c}, Integers,(* failsafe *) Round[n^2/12]], 3]] ]]];
%t A119646 Table[ tanya@n, {n, 4, 108}]
%Y A119646 Cf. A119028.
%Y A119646 Adjacent sequences: A119643 A119644 A119645 this_sequence A119647 A119648 A119649
%Y A119646 Sequence in context: A098398 A131714 A130196 this_sequence A024693 A025126 A129706
%K A119646 nonn
%O A119646 4,10
%A A119646 Joseph Biberstine (jrbibers(AT)indiana.edu) and Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 27 2006
%I A024693
%S A024693 0,1,1,0,1,1,1,2,2,1,2,2,1,2,2,2,2,2,2,2,3,3,2,3,3,3,3,3,2,3,3,3,3,3,2,3,3,
%T A024693 2,3,4,4,4,4,3,4,4,4,4,4,4,4,4,4,3,4,3,4,4,3,4,4,4,4,4,3,4,4,4,4,5,5,5,5,5,
%U A024693 4,5,5,5,5,5,5,5,5,4,5,5,4,5,5,4
%N A024693 s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n+1-k), where k = [ (n+1)/2 ], s = A023533, t = A014306.
%Y A024693 Adjacent sequences: A024690 A024691 A024692 this_sequence A024694 A024695 A024696
%Y A024693 Sequence in context: A131714 A130196 A119646 this_sequence A025126 A129706 A024327
%K A024693 nonn
%O A024693 1,8
%A A024693 Clark Kimberling (ck6(AT)evansville.edu)
%I A025126
%S A025126 1,1,0,1,1,1,2,2,1,2,2,1,2,2,2,2,2,2,2,3,3,2,3,3,3,3,3,2,3,3,3,3,3,2,3,3,2,
%T A025126 3,4,4,4,4,3,4,4,4,4,4,4,4,4,4,3,4,3,4,4,3,4,4,4,4,4,3,4,4,4,4,5,5,5,5,5,4,
%U A025126 5,5,5,5,5,5,5,5,4,5,5,4,5,5,4,5,5,4,5,5,5,5,5,5,5,5,5,4,5,5,5,5,5,5,5,5,6
%N A025126 s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = A023533, t = A014306.
%Y A025126 Adjacent sequences: A025123 A025124 A025125 this_sequence A025127 A025128 A025129
%Y A025126 Sequence in context: A130196 A119646 A024693 this_sequence A129706 A024327 A073044
%K A025126 nonn
%O A025126 1,7
%A A025126 Clark Kimberling (ck6(AT)evansville.edu)
%I A129706
%S A129706 1,2,2,1,2,2,1,2,2,2,2,2,2,2,4,2,1,2,2,2,4,4,4,2,1,2,2,2,4,4,6,6,4,2,2,
%T A129706 2,2,2,4,4,6,8,8,6,6,4,2,1,2,2,2,4,4,6,8,10,10,10,10,8,6,4,2,1,2,2,2,4,
%U A129706 4,6,8,10,12,14,14,14,14,12,10,8,4,2,2,2,2,2,4,4,6,8,10,12,16,18,18,20
%N A129706 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k inversions (n>=0, 0<=k<=floor(n(n+1)/6)). A Fibonacci binary word is a binary word having no 00 subword.
%C A129706 Row n has 1+floor(n(n+1)/6) terms. Row sums are the Fibonacci numbers (A000045). Sum(k*T(n,k), k>=0)=A129707(n).
%F A129706 G.f.=G(t,z)=H(t,1,z), where H(t,x,z)=1+z+xzH(t,x,z)+txz^2*H(t,tx,z). Row generating polynomials P[n] are given by P[n](t)=Q[n](t,1), where Q[0]=1, Q[1]=1+x, Q[n](t,x)=xQ[n-1](t,x)+txQ[n-2](t,tx) for n>=2.
%e A129706 T(5,3)=4 because we have 11101, 10101, 01110, and 01010.
%e A129706 Triangle starts:
%e A129706 1;
%e A129706 2;
%e A129706 2,1;
%e A129706 2,2,1;
%e A129706 2,2,2,2;
%e A129706 2,2,2,4,2,1;
%e A129706 2,2,2,4,4,4,2,1;
%p A129706 Q[0]:=1: Q[1]:=1+x: for n from 2 to 12 do Q[n]:=expand(x*Q[n-1]+t*x*subs(x=t*x,Q[n-2])) od: for n from 0 to 15 do P[n]:=sort(subs(x=1,Q[n])) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..floor(n*(n+1)/6)) od; # yields sequence in triangular form
%Y A129706 Cf. A000045, A129707.
%Y A129706 Adjacent sequences: A129703 A129704 A129705 this_sequence A129707 A129708 A129709
%Y A129706 Sequence in context: A119646 A024693 A025126 this_sequence A024327 A073044 A124800
%K A129706 nonn,tabf
%O A129706 0,2
%A A129706 Emeric Deutsch (deutsch(AT)duke.poly.edu), May 12 2007
%I A024327
%S A024327 0,0,1,1,0,1,1,1,2,2,1,2,2,1,2,2,3,2,3,3,2,3,3,2,3,3,4,3,4,4,4,4,3,4,4,3,4,
%T A024327 4,3,5,5,5,4,5,5,5,5,4,5,5,5,5,6,5,6,6,5,6,6,5,5,6,6,5,6,6,6,6,5,7,7,7,7,7,
%U A024327 6,7,7,7,7,7,7,6,7,7,6,7,8,7,8,7
%N A024327 s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n+1-k), where k=[ (n+1)/2) ], s = A023531, t = 023534.
%Y A024327 Adjacent sequences: A024324 A024325 A024326 this_sequence A024328 A024329 A024330
%Y A024327 Sequence in context: A024693 A025126 A129706 this_sequence A073044 A124800 A069163
%K A024327 nonn
%O A024327 1,9
%A A024327 Clark Kimberling (ck6(AT)evansville.edu)
%I A073044
%S A073044 2,2,1,2,2,1,2,3,2,1,2,4,4,2,1,2,5,6,5,2,1,2,6,9,8,6,2,1,2,7,12,14,10,7,
%T A073044 2,1,2,8,16,20,20,12,8,2,1,2,9,20,30,30,27,14,9,2,1,2,10,25,40,50,42,35,
%U A073044 16,10,2,1,2,11,30,55,70,77,56,44,18,11,2,1,2,12,36,70,105,112,112,72
%N A073044 Triangle read by rows: T(n,k) (n>=1, k>=0) = number of n-sequences of 0's and 1's with no pair of adjacent 0's and exactly k pairs of adjacent 1's.
%D A073044 S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see pp. 67-68).
%F A073044 Recurrence: T(n, k)=T(n-1, k-1)+T(n-2, k).
%F A073044 G.f.=G(t, z)=z(2+2z-tz)/(1-tz-z^2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 01 2005
%e A073044 T(5,2)=4 because the sequences of length 5 with 2 pairs 11 are 11101, 11011,10111, 01110.
%e A073044 Triangle starts:
%e A073044 2;
%e A073044 2,1;
%e A073044 2,2,1;
%e A073044 2,3,2,1;
%e A073044 2,4,4,2,1;
%p A073044 G:=z*(2+2*z-t*z)/(1-t*z-z^2):Gser:=simplify(series(G,z=0,17)):for n from 1 to 15 do P[n]:=sort(coeff(Gser,z^n)) od:for n from 1 to 13 do seq(coeff(t*P[n],t^k),k=1..n) od;# yields sequence in triangular form
%Y A073044 Row sums are the Fibonacci numbers (A000045).
%Y A073044 Adjacent sequences: A073041 A073042 A073043 this_sequence A073045 A073046 A073047
%Y A073044 Sequence in context: A025126 A129706 A024327 this_sequence A124800 A069163 A025260
%K A073044 nonn,tabl
%O A073044 1,1
%A A073044 Roger Cuculiere (cuculier(AT)imaginet.fr), Aug 24 2002
%E A073044 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 01 2005
%I A124800
%S A124800 2,2,1,2,2,1,2,3,3,1,2,4,6,4,3,2,5,10,10,15,9,2,6,15,20,45,54,23,2,7,21,
%T A124800 35,105,189,161,53,2,8,28,56,210,504,644,424,115
%V A124800 2,2,1,2,2,1,2,3,3,-1,2,4,6,-4,3,2,5,10,-10,15,-9,2,6,15,-20,45,-54,23,2,7,21,-35,105,
%W A124800 -189,161,-53,2,8,28,-56,210,-504,644,-424,115
%N A124800 Triangle, row sums = primes.
%C A124800 Right border = A007442, (2, 1, 1, -1, 3, -9...) inverse binomial transform of the primes.
%F A124800 Let M = a diagonalized infinite matrix of A007442, inverse binomial transform of the primes, and P = Pascal's triangle as an infinite lower triangular matrix. Then the triangle A124800 = P*M, deleting the zeros.
%e A124800 First few rows of the triangle are:
%e A124800 2;
%e A124800 2, 1;
%e A124800 2, 2, 1;
%e A124800 2, 3, 3, -1;
%e A124800 2, 4, 6, -4, 3;
%e A124800 2, 5, 10, 10, 15, -9;
%e A124800 2, 6, 15, -20, 45, -54, 23;
%e A124800 2, 7, 21, -35, 105, -189, 161, -53;
%e A124800 ...
%e A124800 Row 5 sum = 11 = p5 since (2 + 4 + 6 - 4 + 3) = 11.
%Y A124800 Cf. A007442.
%Y A124800 Adjacent sequences: A124797 A124798 A124799 this_sequence A124801 A124802 A124803
%Y A124800 Sequence in context: A129706 A024327 A073044 this_sequence A069163 A025260 A123369
%K A124800 tabl,sign
%O A124800 1,1
%A A124800 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 07 2006
%I A069163
%S A069163 0,1,2,2,1,2,2,1,2,4,1,2,3,1,5,3,1,4,4,3,5,3,2,4,4,1,4,4,2,5,4,0,6,2,3,
%T A069163 4,4,2,4,8,0,3,4,2,5,4,4,5,5,3,7,5,3,5,7,2,4,6,3,7,7,5,6,6,5,5,7,5,6,8,
%U A069163 1,3,8,3,11,6,1,10,5,2,5,8,5,5,7,5,4,6,2,8,7,4,13,7,5,9,7,4,9
%N A069163 Number of symmetric primes between n^2 and (n+2)^2. Two primes are termed symmetric in n^2 to (n+2)^2 if there is a k < 2n such that mid-k and mid+k are both prime, where mid =n(n+2).
%C A069163 This relates primes between n^2 and (n+1)^2 to primes between (n+1)^2 and (n+2)^2. It appears that the number of symmetric primes is zero for only n=0,32,41.
%e A069163 a(5) = 1 because in the range 25 to 49, the primes 29 and 41 are the only primes symmetric about the number 35.
%t A069163 maxN=100; lst={}; For[n=1, n= 4.
%Y A025260 Adjacent sequences: A025257 A025258 A025259 this_sequence A025261 A025262 A025263
%Y A025260 Sequence in context: A073044 A124800 A069163 this_sequence A123369 A023671 A117535
%K A025260 sign
%O A025260 1,1
%A A025260 Clark Kimberling (ck6(AT)evansville.edu)
%I A123369
%S A123369 0,1,1,2,2,1,2,2,1,3,2,2,2,3,2,4,3,1,2,5,3,3,2,2,1,3
%N A123369 Number of prime divisors of n-th Conway and Guy second-order harmonic number (counted with multiplicity).
%C A123369 We must include multiplicity in the definition due to terms such as a(16) = 29889983 = 19 * 31^2 * 1637. The primes are those n for which a(n) = Omega(A027612(n))= 1, namely a(2) = 5, a(3) = 13, a(6) = 223, a(9) = 4861, a(18) = 197698279, a(25) = 25472027467. The semiprimes are those for which a(n) = 2, such as when n = 4, 5, 7, 8, 11, 12, 13, 15, 19, 23, 24. The 3-almost primes are those for which a(n) = 3, as with the "3-brilliant" a(10) = 55991 = 13 * 59 * 73, a(14), a(17), a(21), a(22), a(26).
%D A123369 J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, pp. 143 and 258-259.
%H A123369 Eric Weisstein's World of Mathematics, Harmonic Number, MathWorld, see discussion of Conway and Guy (1996) definition of the second-order harmonic number.
%F A123369 a(n) = A001222(A027612(n)) = Omega(Numerator of 1/n + 2/(n-1) + 3/(n-2) +...+ (n-1)/2 + n).
%e A123369 a(20) = 5 because A027612(20) = 41054655 = 3 * 5 * 23 * 127 * 937 has 5 prime factors.
%Y A123369 Cf. A001222 Number of prime divisors of n (counted with multiplicity), A027612 Numerator of 1/n + 2/(n-1) + 3/(n-2) +...+ (n-1)/2 + n, A027611, A001008, A002805, A001705, A006675, A093418.
%Y A123369 Adjacent sequences: A123366 A123367 A123368 this_sequence A123370 A123371 A123372
%Y A123369 Sequence in context: A124800 A069163 A025260 this_sequence A023671 A117535 A072463
%K A123369 easy,nonn
%O A123369 1,4
%A A123369 Jonathan Vos Post (jvospost2(AT)yahoo.com), Nov 09 2006
%I A023671
%S A023671 0,1,1,0,2,2,1,2,2,1,3,3,1,3,3,3,3,3,2,2,4,4,2,4,4,4,4,4,2,4,4,4,4,
%T A023671 4,3,5,5,3,4,5,5,5,5,3,5,5,5,5,5,5,5,5,5,3,5,4,6,6,4,6,6,6,6,6,4,6,
%U A023671 6,6,5,6,6,6,6,6,4,6,6,6,6,6,6,6,6,5,7,7,5,7,7,5
%N A023671 Convolution of A023533 and A014306.
%Y A023671 Adjacent sequences: A023668 A023669 A023670 this_sequence A023672 A023673 A023674
%Y A023671 Sequence in context: A069163 A025260 A123369 this_sequence A117535 A072463 A128853
%K A023671 nonn
%O A023671 1,5
%A A023671 Clark Kimberling (ck6(AT)evansville.edu)
%I A117535
%S A117535 1,1,2,2,1,2,2,1,3,3,2,4,4,2,3,3,1,3,3,2,4,4,2,3,3,1,4,4,3,6,6,3,5,5,2,
%T A117535 6,6,4,8,8,4,6,6,2,5,5,3,6,6,3,4,4,1,4,4,3,6,6,3,5,5,2,6,6,4,8,8,4,6,6,
%U A117535 2,5,5,3,6,6,3,4,4,1,5,5,4,8,8,4,7,7,3,9,9,6,12,12,6,9,9,3,8,8,5,10,10
%N A117535 Number of ways of writing n as a sum of powers of 3, each power being used at most 4 times.
%C A117535 It seems that this sequence can be calculated by constructing an insertion tree in which the insertion rules depend on the "age" of a term at a particular stage of the calculation. See the link for a discussion of this concept.
%H A117535 John W. Layman, Sequences Generated by Age-Determined Insertion Trees
%F A117535 G.f.=-1+product((1+x^(3^j)+x^(2*(3^j))+x^(3*(3^j))+x^(4*(3^j))), j=0..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 02 2006
%e A117535 a(12)=4 because 12=9+3=9+1+1+1=3+3+3+3=3+3+3+1+1+1.
%p A117535 g:=-1+product((1+x^(3^j)+x^(2*(3^j))+x^(3*(3^j))+x^(4*(3^j))),j=0..10): gser:=series(g,x=0,106): seq(coeff(gser,x,n),n=1..103); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 02 2006
%Y A117535 Cf. A054390.
%Y A117535 Adjacent sequences: A117532 A117533 A117534 this_sequence A117536 A117537 A117538
%Y A117535 Sequence in context: A025260 A123369 A023671 this_sequence A072463 A128853 A136165
%K A117535 nonn
%O A117535 1,3
%A A117535 John W. Layman (layman(AT)math.vt.edu), Mar 27 2006
%I A072463
%S A072463 0,1,1,2,2,1,2,2,2,1,1,1,3,2,3,2,1,1,3,1,2,2,1,1,4,1,1,1,2,1,3,3,3,1,1,
%T A072463 1,3,1,2,2,2,1,4,1,2,2,1,1,5,1,1,1,1,1,3,1,3,2,1,1,6,1,3,2,1,1,1,1,2,1,
%U A072463 1,1,8,1,2,1,1,1,2,1,3,1,1,1,6,1,1,1,1,1,5,2,1,3,1,1,5,1,3,1,1,1,2,1,3
%N A072463 Shadow transform of sigma(n), A000203, starting with sigma(0), sigma(1), ...
%H A072463 N. J. A. Sloane, Transforms
%Y A072463 Adjacent sequences: A072460 A072461 A072462 this_sequence A072464 A072465 A072466
%Y A072463 Sequence in context: A123369 A023671 A117535 this_sequence A128853 A136165 A134193
%K A072463 nonn
%O A072463 0,4
%A A072463 njas, Aug 02 2002
%I A128853
%S A128853 1,2,2,1,2,2,2,1,1,2,2,2,2,2,4,1,2,1,2,2,2,2,2,2,1,2,1,2,2,4,2,1,4,2,4,
%T A128853 1,2,2,2,2,2,2,2,2,2,2,2,2,1,1,4,2,2,1,2,2,2,2,2,4,2,2,2,1,4,4,2,2,4,4,
%U A128853 2,1,2,2,2,2,4,2,2,2,1,2,2,2,4,2,4,2,2,2,4,2,2,2,4,2,2,1,2,1,2,4,2,2,4
%N A128853 a(n) = the number of positive divisors of n which are coprime to phi(n) = A000010(n).
%e A128853 12 is coprime to 4 positive integers (1,5,7 and 11) which are <= 12; so phi(12)=4. There are 2 divisors (1 and 3) of 12 that are coprime to 4. So a(12) = 2.
%p A128853 with(numtheory): a:=proc(n) local div,ct,j: div:=divisors(n): ct:=0: for j from 1 to tau(n) do if igcd(div[j],phi(n))=1 then ct:=ct+1 else fi od: ct; end: seq(a(n),n=1..140); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 17 2007
%Y A128853 Adjacent sequences: A128850 A128851 A128852 this_sequence A128854 A128855 A128856
%Y A128853 Sequence in context: A023671 A117535 A072463 this_sequence A136165 A134193 A085030
%K A128853 nonn
%O A128853 1,2
%A A128853 Leroy Quet (qq-quet(AT)mindspring.com), Apr 16 2007
%E A128853 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 17 2007
%I A136165
%S A136165 1,2,2,1,2,2,2,1,1,2,2,3,2,2,2,1,2,2,2,1,2,2,2,3,1,2,1,1,2,4
%N A136165 a(n) = the number of divisors of n that are each not noncoprime to either the previous or the following divisor of n.
%e A136165 The divisors of 28 are 1,2,4,7,14,28. Among adjacent pairs of divisors, 1 and 2 are coprime, and 4 and 7 are coprime. But 2 and 4 are adjacent divisors which are not coprime. And 7 is next to 14 among divisors of 28, and 7 and 14 are not coprime. Therefore there is a(28) = 1 divisor (1) that is not next to another divisor that it is not coprime to.
%Y A136165 Cf. A136164, A136166, A136167.
%Y A136165 Adjacent sequences: A136162 A136163 A136164 this_sequence A136166 A136167 A136168
%Y A136165 Sequence in context: A117535 A072463 A128853 this_sequence A134193 A085030 A078377
%K A136165 more,nonn
%O A136165 1,2
%A A136165 Leroy Quet (qq-quet(AT)mindspring.com), Dec 17 2007
%I A134193
%S A134193 1,2,2,1,2,2,2,1,1,2,2,3,2,2,2,1,2,3,2,3,2,2,2,2,1,2,1,3,2,2,2,1,2,2,2,1,
%T A134193 2,2,2,2,2,2,2,3,3,2,2,2,1,3,2,3,2,2,2,2,2,2,2,3,2,2,3,1,2,2,2,3,2,2,2,
%U A134193 1,2,2,3,3,2,2,2,2,1,2,2,3,2,2,2,2,2,3,2,3,2,2,2,2,2,3,3,1,2,2,2,2,2,2
%N A134193 a(1) = 1; for n>1, a(n) = the smallest positive integer not occurring among the exponents in the prime-factorization of n.
%e A134193 The prime factorization of 24 is 2^3 * 3^1. The exponents are 3 and 1. Therefore a(24) = 2 is the smallest positive integer not occurring among (3,1).
%t A134193 Table[Complement[Range[n], Table[FactorInteger[n][[i, 2]], {i, 1, Length[FactorInteger[n]]}]][[1]], {n, 2, 120}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Jan 21 2008
%Y A134193 Adjacent sequences: A134190 A134191 A134192 this_sequence A134194 A134195 A134196
%Y A134193 Sequence in context: A072463 A128853 A136165 this_sequence A085030 A078377 A105697
%K A134193 nonn
%O A134193 1,2
%A A134193 Leroy Quet (qq-quet(AT)mindspring.com), Jan 13 2008
%E A134193 More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Jan 21 2008
%I A085030
%S A085030 2,2,1,2,2,2,1,2,2,1,1,1,1,2,2,3,2,2,3,2,2,3,2,1,4,2,4,1,3,2,2,3,2,2,4,
%T A085030 2,3,3,2,2,2,4,2,2,3,2,1,1,1,2,3,3,3,4,5,1,1,2,2,2,3,3,3,4,3,2,5,4,4,2,
%U A085030 2,3,3,3,5,2,3,2,3,2,2,4,3,2,3,4,3,1,6,1,2,1,4,3,4,2,3,3,4,3
%N A085030 Number of prime factors of cyclotomic(n,5), which is A019323(n), the value of the n-th cyclotomic polynomial evaluated at x=5.
%C A085030 The Mobius transform of this sequence yields A057956, number of prime factors of 5^n-1.
%D A085030 See references at A085021
%t A085030 Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 5]]][[2]], {n, 1, 100}]
%Y A085030 Cf. A019323, A057956, A085021.
%Y A085030 Adjacent sequences: A085027 A085028 A085029 this_sequence A085031 A085032 A085033
%Y A085030 Sequence in context: A128853 A136165 A134193 this_sequence A078377 A105697 A080757
%K A085030 nonn
%O A085030 1,1
%A A085030 T. D. Noe (noe(AT)sspectra.com), Jun 19 2003
%I A078377
%S A078377 1,1,1,2,2,1,2,2,2,1,2,2,1,1,2,2,2,2,2,2,2,2,1,2,2,2,1,2,2,1,2,1,2,3,1,
%T A078377 1,2,2,2,2,2,3,2,2,2,3,2,3,2,2,2,2,1,3,2,2,2,3,1,2,2,2,1,1,3,3,1,3,3,2,
%U A078377 1,3,1,1,3,3,2,2,2,2,2,2,1,2,1,3,2,2,1,1,3,2,2,2,1,2,3,2,2,1,2,3,2,3,2
%N A078377 Number of distinct prime factors of numbers which can be written as sum of a positive square and a positive cube.
%C A078377 a(n)=A001221(A055394(n)).
%Y A078377 Cf. A078378, A078379.
%Y A078377 Adjacent sequences: A078374 A078375 A078376 this_sequence A078378 A078379 A078380
%Y A078377 Sequence in context: A136165 A134193 A085030 this_sequence A105697 A080757 A037196
%K A078377 nonn
%O A078377 1,4
%A A078377 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Nov 25 2002
%I A105697
%S A105697 2,2,1,2,2,2,1,2,2,1,2,2,2,1,2,2,2,1,2,2,1,2,2,2,2,1,1,2,2,2,2,1,2,2,2,
%T A105697 2,1,2,2,1,1,2,2,2,2,2,2,1,1,2,2,2,2,2,2,1,1,2,2,2,2,1,1,2,2,2,2,2,1,2,
%U A105697 1,2,2,2,1,2,1,2,2,2,2,2
%N A105697 A simple "Fractal Jump Sequence" (FJS). A FJS is a sequence of digits embedding an infinite amount of copies of itself. Modus operandi: underline the first digit "a" of such a sequence then jump over the next "a" digits and underline the digit "b" on which you land. Jump now from there over the next "b" digits and underline the digit "c" on which you land. Etc. The "abc...n..." succession of underlined digits is the sequence itself.
%C A105697 This is how to construct the sequence: start with 2 on rows a and b; put 2 empty spaces behind the 2 on row a; choose any two digits and put them on row b under the 2 empty spaces of row a; go back to row a and add the same two digits but each one with its according spaces (1 must always be followed by 1 space on row a and 2 must always be followed by 2 spaces); go back to row b and add under the next available spaces of a the digits necessary so to have the same succession of digits in rows b and a. The sequence builds itself automatically. The row (c) is obtained by "pushing" (a) into (b) -- [the first digit of a and b melt in a single copy of themselves]. Row (c) is the FJS sequence above.
%C A105697 (a)..2..2..1.2..2..2..1.2..2..1.2..2..2..1.2
%C A105697 (b)..221.22.2.12.21.22.2.12.22.1.22.12.22.2.1
%C A105697 .....----------------------------------------
%C A105697 (c)..2212221221222122212212222112222122221221
%e A105697 To build such sequences one has only to choose the first digit d and the d digits to put under the d spaces of row (a).
%Y A105697 Adjacent sequences: A105694 A105695 A105696 this_sequence A105698 A105699 A105700
%Y A105697 Sequence in context: A134193 A085030 A078377 this_sequence A080757 A037196 A116543
%K A105697 base,easy,nonn,uned
%O A105697 2,1
%A A105697 Eric Angelini (eric.angelini(AT)kntv.be), May 04 2005
%I A080757
%S A080757 1,2,2,1,2,2,2,1,2,2,2,1,2,2,1,2,2,2,1,2,2,2,1,2,2,2,1,2,2,1,2,2,2,1,2,2,
%T A080757 2,1,2,2,2,1,2,2,1,2,2,2,1,2,2,2,1,2,2,1,2,2,2,1,2,2,2,1,2,2,2,1,2,2,1,
%U A080757 2,2,2,1,2,2,2,1,2,2,2,1,2,2,1,2,2,2,1,2,2,2,1,2,2,2,1,2,2,1,2,2,2,1,2
%N A080757 First differences of Beatty sequence A022838(n) = floor(n sqrt(3)).
%F A080757 Let S(0) = 1; obtain S(k) from S(k-1) by applying 1 -> 221, 2 -> 2221; sequence is S(0), S(1), S(2), ... - Matthew Vandermast (ghodges14(AT)comcast.net), Mar 25 2003
%t A080757 Flatten[ NestList[ Flatten[ # /. {1 -> {2, 2, 1}, 2 -> {2, 2, 2, 1}}] &, {1}, 4]] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 20 2005)
%Y A080757 Equals A007538(n+1) - 1. Cf. A001030.
%Y A080757 Adjacent sequences: A080754 A080755 A080756 this_sequence A080758 A080759 A080760
%Y A080757 Sequence in context: A085030 A078377 A105697 this_sequence A037196 A116543 A107260
%K A080757 nonn
%O A080757 0,2
%A A080757 njas, Mar 25 2003
%I A037196
%S A037196 2,2,1,2,2,2,1,2,2,2,1,3,2,3,4,3,3,4,4,3,1,3,2,3,3,3,2,3,3,3,1,3,2,3,3,
%T A037196 3,2,3,3,3,1,3,2,3,3,3,2,3,3,3,1,3,2,3,3,3,2,3,3,3,1,3,2,3,3,3,2,3,3,3,
%U A037196 2,4,3,4,4,4,3,4,4,4,2,4,3,4,4,4,3,4,4,4,2,4,3,4,4,4,3,4,4,4,4,6,5,6,6
%N A037196 Number of vowels in n.
%Y A037196 Adjacent sequences: A037193 A037194 A037195 this_sequence A037197 A037198 A037199
%Y A037196 Sequence in context: A078377 A105697 A080757 this_sequence A116543 A107260 A116204
%K A037196 nonn,word,easy
%O A037196 0,1
%A A037196 njas
%E A037196 More terms from Larry Reeves (larryr(AT)acm.org), Sep 25 2000
%I A116543
%S A116543 1,1,1,1,2,2,1,2,2,2,1,2,2,2,2,3,3,1,2,2,2,2,3,3,2,3,3,3,1,2,2,2,2,3,3,
%T A116543 2,3,3,3,2,3,3,3,3,4,4,1,2,2,2,2,3,3,2,3,3,3,2,3,3,3,3,4,4,2,3,3,3,3,4,
%U A116543 4,3,4,4,4,1,2,2,2,2,3,3,2,3,3,3,2,3,3,3,3,4,4,2,3,3,3,3,4,4,3,4,4,4,2
%N A116543 Number of terms in greedy representation of n of the Lucas numbers.
%C A116543 I have been researching A007895 and similar sequences, and created this sequence as an analogue of A007895 for the Lucas sequence (A000032).
%H A116543 Ron Knott, Using the Fibonacci numbers to represent whole numbers.
%F A116543 Let L(N)=max(Lucas numbers < N). Then a(0)=0 a(N)=1+a(N-L(N)).
%e A116543 a(12)=2 because 12=11+1
%Y A116543 Cf. A131343, A000032, A007895.
%Y A116543 Adjacent sequences: A116540 A116541 A116542 this_sequence A116544 A116545 A116546
%Y A116543 Sequence in context: A105697 A080757 A037196 this_sequence A107260 A116204 A106054
%K A116543 nonn
%O A116543 1,5
%A A116543 James Davis (math-man(AT)tamu.edu), Mar 28 2006, Jun 07 2006
%E A116543 Edited by njas, Aug 10 2007
%I A107260
%S A107260 1,1,1,1,2,2,1,2,2,2,1,3,2,2,2,3,3,1,3,2,2,2,3,3,2,3,3,3,1,4,2,2,2,3,3,
%T A107260 2,3,3,3,2,4,3,3,3,4,4,1,4,2,2,2,3,3,2,3,3,3,2,4,3,3,3,4,4,2,4,3,3,3,4,
%U A107260 4,3,4,4,4,1,5,2,2,2,3,3,2,3,3,3,2,4,3,3,3,4,4,2,4,3,3,3,4,4,3,4
%N A107260 Number of Lucas numbers needed to sum up to n.
%C A107260 Cf. A007895. Record positions are in A002878 + 1.
%Y A107260 Adjacent sequences: A107257 A107258 A107259 this_sequence A107261 A107262 A107263
%Y A107260 Sequence in context: A080757 A037196 A116543 this_sequence A116204 A106054 A023568
%K A107260 nonn
%O A107260 2,5
%A A107260 Ralf Stephan (ralf(AT)ark.in-berlin.de), May 15 2005
%I A116204
%S A116204 1,1,2,2,1,2,2,2,1,3,4,2,2,2,2,4,1,2,3,2,2,4,2,2,2,3,4,4,2,2,4,2,1,4,2,
%T A116204 4,3,2,2,4,2,2,4,2,2,6,2,2,2,3,6,2,2,2,4,4,2,4,2,2,4,2,2,6,1,4,4,2,2,4,
%U A116204 4,2,3,2,2,6,2,4,4,2,2,5,4,2,4,4,2,4,2,2,6,4,2,4,2,4,2,2,3,2,3,2,4,2,2
%N A116204 a(0) = 1; for n>=1, a(n) = the number of positive divisors of n which are coprime to a(n-1).
%e A116204 a(11) = 2. There are 2 positive divisors (1 and 3) of 12 which are coprime to 2. So a(12) = 2.
%p A116204 with(numtheory): a[0]:=1: for n from 1 to 140 do ct:=0: div:=divisors(n): for j from 1 to tau(n) do if igcd(div[j],a[n-1])=1 then ct:=ct+1 else ct:=ct: fi: od: a[n]:=ct: od: seq(a[n],n=0..140); (Deutsch)
%p A116204 A116204 := proc(nmax) local a,n,dvs,resl,d ; a := [1] ; while nops(a) < nmax do n := nops(a) ; dvs := numtheory[divisors](n) ; resl :=0 ; for d from 1 to nops(dvs) do if gcd(op(d,dvs), op(-1,a)) = 1 then resl := resl+1 ; fi ; od ; a := [op(a),resl] ; od ; RETURN(a) ; end: A116204(100) ; (Mathar)
%Y A116204 Adjacent sequences: A116201 A116202 A116203 this_sequence A116205 A116206 A116207
%Y A116204 Sequence in context: A037196 A116543 A107260 this_sequence A106054 A023568 A081753
%K A116204 nonn
%O A116204 0,3
%A A116204 Leroy Quet (qq-quet(AT)mindspring.com), Apr 16 2007
%E A116204 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl) and Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 27 2007
%I A106054
%S A106054 2,2,1,2,2,2,2,1,2,2,1,1,4,4,3,4,4,1,1,2,2,1,2,2,2,2,1,2,2,2,2,1,2,
%T A106054 2,2,2,1,2,2,1,1,4,4,3,4,4,1,1,2,2,1,2,2,2,2,1,2,2,1,1,4,4,3,4,4,1,
%U A106054 1,1,1,4,4,3,4,4,1,1,4,4,3,4,4,4,4,3,4,4,3,3,2,2,1,2,2,3,3,4,4,3,4
%N A106054 Trajectory of 1 under the morphism 1->{2,2,1,2,2}, 2->{3}, 3->{4,4,3,4,4}, 4->{1}.
%C A106054 Pentasilver dragon 5-symbol substitution, characteristic polynomial x^4-2*x^3+x-16.
%C A106054 The existence of the three polynomials silver: x^4-2*x^3+x^2-4, double silver: x^4-4x^3+4x^2-4 and pentasilver: x^4-2*x^3+x-16 confirms that a Kenyon-like polynomial of a general form: x^4-p*x^3+q*x^2-r exists with substitutionms associated to it.
%t A106054 s[1] = {2, 2, 1, 2, 2}; s[2] = {3}; s[3] = {4, 4, 3, 4, 4}; s[4] = {1}; t[a_] := Flatten[s /@ a]; p[0] = {1}; p[1] = t[p[0]]; p[n_] := t[p[n - 1]] aa = p[5]
%Y A106054 Adjacent sequences: A106051 A106052 A106053 this_sequence A106055 A106056 A106057
%Y A106054 Sequence in context: A116543 A107260 A116204 this_sequence A023568 A081753 A089049
%K A106054 nonn
%O A106054 0,1
%A A106054 Roger Bagula (rlbagulatftn(AT)yahoo.com), May 06 2005
%E A106054 Corrected and edited by njas, Jun 03 2005
%I A023568
%S A023568 0,0,1,1,1,2,2,1,2,2,2,2,2,2,2,2,2,2,1,2,3,2,2,2,2,2,2,2,2,3,2,1,2,
%T A023568 2,2,2,3,2,2,3,2,2,2,3,2,2,2,3,2,2,3,2,3,2,2,3,3,2,2,2,3,3,2,3,3,2,
%U A023568 2,2,2,2,3,2,3,3,2,3,2,2,2,3,2,3,2,3,2,3,2,2,2,3,2,3,2,2,2,2,3,3,3
%N A023568 Number of distinct prime divisors of prime(n)-3.
%Y A023568 Adjacent sequences: A023565 A023566 A023567 this_sequence A023569 A023570 A023571
%Y A023568 Sequence in context: A107260 A116204 A106054 this_sequence A081753 A089049 A029420
%K A023568 nonn
%O A023568 1,6
%A A023568 Clark Kimberling (ck6(AT)evansville.edu)
%I A081753
%S A081753 1,1,0,1,1,1,1,1,1,1,1,1,2,2,1,2,2,2,2,2,2,2,2,2,3,3,2,3,3,3,3,3,3,3,3,
%T A081753 3,4,4,3,4,4,4,4,4,4,4,4,4,5,5,4,5,5,5,5,5,5,5,5,5,6,6,5,6,6,6,6,6,6,6,
%U A081753 6,6,7,7,6,7,7,7,7,7,7,7,7,7,8,8,7,8,8,8,8,8,8,8,8,8,9,9,8,9,9,9,9,9,9
%N A081753 a(n) = floor(n/12) if n==2 (mod 12); a(n)=floor(n/12)+1 otherwise.
%C A081753 a(2n)=dimension of M(2n), where M(2n) denotes the complex vector space of modular forms of weight 2n for the group : PSL2(Z). dimension of M(2n+1)=0.
%D A081753 Yves Hellegouarch, "Invitation aux mathematiques de Fermat-Wiles", Dunod, 2eme edition, p. 285
%o A081753 (PARI) a(k)=if(k%12-2,floor(k/12)+1,floor(k/12))
%Y A081753 Cf. A008615.
%Y A081753 Adjacent sequences: A081750 A081751 A081752 this_sequence A081754 A081755 A081756
%Y A081753 Sequence in context: A116204 A106054 A023568 this_sequence A089049 A029420 A029405
%K A081753 nonn
%O A081753 0,13
%A A081753 Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 08 2003
%I A089049
%S A089049 0,0,0,0,1,1,1,2,2,1,2,2,2,2,3,1,2,1,2,2,3,2,3,1,2,2,3,1,3,1,1,0,2,1,2,
%T A089049 2,3,2,3,1,3,2,3,1,3,1,1,0,2,2,3,1,3,1,1,0,3,1,1,0,1,0,0,0,2,1,2,2,3,2,
%U A089049 3,1,3,2,3,1,3,1,1,0,3,2,3,1,3,1,1,0,3,1,1,0,1,0,0,0,2,2,3,1,3,1,1,0,3
%N A089049 Number of ways of writing n as a sum of exactly 4 powers of 2.
%C A089049 The powers do not need to be distinct.
%Y A089049 A column of A089052. Cf. A036987, A075897, A089048, A089050, A089051, A089053.
%Y A089049 Adjacent sequences: A089046 A089047 A089048 this_sequence A089050 A089051 A089052
%Y A089049 Sequence in context: A106054 A023568 A081753 this_sequence A029420 A029405 A029350
%K A089049 nonn
%O A089049 0,8
%A A089049 njas, Dec 03 2003
%I A029420
%S A029420 1,0,0,0,0,0,1,1,0,0,1,1,1,1,1,0,1,2,2,1,2,2,2,2,3,2,2,
%T A029420 3,4,3,4,4,4,4,5,5,5,5,6,6,7,7,8,7,8,8,9,9,10,10,11,11,
%U A029420 12,12,13,13,14,14,15,15,17,17,18,18,19,19,21,21,22,22
%N A029420 Expansion of 1/((1-x^6)(1-x^7)(1-x^10)(1-x^11)).
%Y A029420 Adjacent sequences: A029417 A029418 A029419 this_sequence A029421 A029422 A029423
%Y A029420 Sequence in context: A023568 A081753 A089049 this_sequence A029405 A029350 A000003
%K A029420 nonn
%O A029420 0,18
%A A029420 njas
%I A029405
%S A029405 1,0,0,0,0,1,0,0,1,1,1,0,1,1,1,1,1,2,2,1,2,2,2,2,3,3,3,
%T A029405 3,3,4,4,3,5,5,5,5,6,6,6,6,7,8,8,7,9,10,9,9,11,11,12,11,
%U A029405 12,14,14,13,15,16,16,16,18,18,19,19,20,22,22,21,24,25
%N A029405 Expansion of 1/((1-x^5)(1-x^8)(1-x^9)(1-x^12)).
%Y A029405 Adjacent sequences: A029402 A029403 A029404 this_sequence A029406 A029407 A029408
%Y A029405 Sequence in context: A081753 A089049 A029420 this_sequence A029350 A000003 A029395
%K A029405 nonn
%O A029405 0,18
%A A029405 njas
%I A029350
%S A029350 1,0,0,0,1,0,1,1,1,0,1,2,2,1,2,2,2,2,4,3,3,3,5,4,5,5,6,
%T A029350 5,7,7,8,7,9,9,10,10,12,11,12,13,15,14,16,16,18,17,20,20,
%U A029350 22,21,24,24,26,26,29,29,31,31,34,34,37,37,40,40,43,43
%N A029350 Expansion of 1/((1-x^4)(1-x^6)(1-x^7)(1-x^11)).
%Y A029350 Adjacent sequences: A029347 A029348 A029349 this_sequence A029351 A029352 A029353
%Y A029350 Sequence in context: A089049 A029420 A029405 this_sequence A000003 A029395 A029282
%K A029350 nonn
%O A029350 0,12
%A A029350 njas
%I A000003 M0196 N0073
%S A000003 1,1,1,1,2,2,1,2,2,2,3,2,2,4,2,2,4,2,3,4,4,2,3,4,2,6,3,2,6,4,3,4,4,4,
%T A000003 6,4,2,6,4,4,8,4,3,6,4,4,5,4,4,6,6,4,6,6,4,8,4,2,9,4,6,8,4,4,8,8,3,8,
%U A000003 8,4,7,4,4,10,6,6,8,4,5,8,6,4,9,8,4,10,6,4,12,8,6,6,4,8,8,8,4,8,6,4
%N A000003 Number of classes of primitive binary forms of discriminant D = -4n; or equivalently class number of quadratic order of discriminant D = -4n.
%D A000003 H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, p. 514.
%D A000003 Fell, Harriet; Newman, Morris; Ordman, Edward; Tables of genera of groups of linear fractional transformations. J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.
%D A000003 D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699.
%H A000003 N. J. A. Sloane, Table of n, a(n) for n = 1..5000
%o A000003 (MAGMA) O1 := MaximalOrder(QuadraticField(D)); _,f := IsSquare(D div Discriminant(O1)); ClassNumber(sub);
%o A000003 (PARI) a(n)=qfbclassno(-4*n)
%Y A000003 Adjacent sequences: A000001 A000002 this_sequence A000004 A000005 A000006
%Y A000003 Sequence in context: A029420 A029405 A029350 this_sequence A029395 A029282 A029286
%K A000003 nonn,nice,easy
%O A000003 1,5
%A A000003 njas
%I A029395
%S A029395 1,0,0,0,0,1,0,1,1,0,1,1,1,1,1,2,2,1,2,2,2,3,3,3,3,3,4,
%T A029395 4,4,5,5,5,6,6,6,7,7,8,8,8,10,9,10,11,11,12,12,13,14,14,
%U A029395 15,16,16,17,18,19,20,20,21,22,23,24,25,26,27,28,29,30
%N A029395 Expansion of 1/((1-x^5)(1-x^7)(1-x^8)(1-x^11)).
%Y A029395 Adjacent sequences: A029392 A029393 A029394 this_sequence A029396 A029397 A029398
%Y A029395 Sequence in context: A029405 A029350 A000003 this_sequence A029282 A029286 A050333
%K A029395 nonn
%O A029395 0,16
%A A029395 njas
%I A029282
%S A029282 1,0,0,1,0,1,1,0,2,2,1,2,2,2,3,3,3,4,5,4,5,6,5,7,8,7,9,
%T A029282 10,9,11,12,11,14,15,14,17,18,17,20,21,21,24,25,25,28,30,
%U A029282 29,32,35,34,38,40,39,44,46,45,50,52,52,57,59,59,64,67
%N A029282 Expansion of 1/((1-x^3)(1-x^5)(1-x^8)(1-x^9)).
%Y A029282 Adjacent sequences: A029279 A029280 A029281 this_sequence A029283 A029284 A029285
%Y A029282 Sequence in context: A029350 A000003 A029395 this_sequence A029286 A050333 A137419
%K A029282 nonn
%O A029282 0,9
%A A029282 njas
%I A029286
%S A029286 1,0,0,1,0,1,1,0,1,2,2,1,2,2,2,4,2,2,5,4,5,5,4,6,7,7,6,
%T A029286 8,9,9,12,9,10,14,12,14,15,14,17,19,19,18,21,22,22,27,24,
%U A029286 25,31,29,32,33,32,36,39,40,39,43,45,46,52,48,50,58,56
%N A029286 Expansion of 1/((1-x^3)(1-x^5)(1-x^9)(1-x^10)).
%Y A029286 Adjacent sequences: A029283 A029284 A029285 this_sequence A029287 A029288 A029289
%Y A029286 Sequence in context: A000003 A029395 A029282 this_sequence A050333 A137419 A057536
%K A029286 nonn
%O A029286 0,10
%A A029286 njas
%I A050333
%S A050333 1,1,0,1,1,1,1,2,2,1,2,2,3,1,2,3,4,1,3,4,3,5,4,6,2,4,7,4,7,5,8,2,6,10,
%T A050333 5,12,7,7,10,2,12,7,13,8,14,6,18,10,13,13,2,17,8,12,23,11,20,7,25,12,
%U A050333 20,16,3,26,11,22,34,14,22,25,8,26,28,35,16,30,39,21,15,3,35,14,39
%N A050333 Number of factorizations into distinct numbers with an odd number of prime factors indexed by prime signatures. A050332(A025487).
%Y A050333 Adjacent sequences: A050330 A050331 A050332 this_sequence A050334 A050335 A050336
%Y A050333 Sequence in context: A029395 A029282 A029286 this_sequence A137419 A057536 A014420
%K A050333 nonn
%O A050333 1,8
%A A050333 Christian G. Bower (bowerc(AT)usa.net), Oct 15 1999.
%I A137419
%S A137419 1,2,2,1,2,2,3,2,3,3,4,3,3,4,5,3,4,5,5,4,5,5,6,5,6,6,7,6,8,7,8,8,9,8,10,
%T A137419 9,10,9,11,9,11,9,12,10,12,10,13,11,12,11,13,12,13,12,14,12,14,12,15,13,
%U A137419 15,13,16,14,15,14,18,15,16,15,19,16,17,15,20,18,18,16,21,19,19,17,23
%N A137419 a(n) = (a(n - 4) + 1 - ((-1)^a(a(n - 1)) + 1)*(a(a(n - 1)) - a(a(n - 2)))/2).
%C A137419 A recursive chaotic sequence based on Bertran Steinsky's sequence.
%C A137419 I started with the sequence from Bertran Steinsky quoted in A000002:
%C A137419 a[1] = 1; a[2] = 2; a[3] = 2;
%C A137419 a[n_] := a[n] = a[n - 1] + 1 - ((-1)^a[a[n - 1]] + 1)*(a[n - 1] - a[n - 2])/2;
%C A137419 and worked to minimize it while getting a more chaotic result.
%D A137419 Bertran Steinsky, A Recursive Formula for the Kolakoski Sequence A000002, J. Integer Sequences, Vol. 9 (2006), Article 06.3.7.
%F A137419 a(1) = 1; a(2) = 2; a(3) = 2; a(4) = 1; a(n) = (a(n - 4) + 1 - ((-1)^a(a(n - 1)) + 1)*(a(a(n - 1)) - a(a(n - 2)))/2)
%t A137419 Clear[a] a[1] = 1; a[2] = 2; a[3] = 2; a[4] = 1; a[n_] := a[n] = (a[n - 4] + 1 - ((-1)^a[a[n - 1]] + 1)*(a[a[n - 1]] - a[a[n - 2]])/2); Table[a[n], {n, 1, 100}]
%Y A137419 Cf. A000002.
%Y A137419 Adjacent sequences: A137416 A137417 A137418 this_sequence A137420 A137421 A137422
%Y A137419 Sequence in context: A029282 A029286