A151552 -id:A151552 - cp's OEIS frontend

A151694 G.f.: Product_{k>=1} (1 + 2x^(2^k-1) + 2x^(2^k)).

1, 2, 2, 2, 6, 8, 4, 2, 6, 8, 8, 16, 28, 24, 8, 2, 6, 8, 8, 16, 28, 24, 12, 16, 28, 32, 48, 88, 104, 64, 16, 2, 6, 8, 8, 16, 28, 24, 12, 16, 28, 32, 48, 88, 104, 64, 20, 16, 28, 32, 48, 88, 104, 72, 56, 88, 120, 160, 272, 384, 336, 160, 32, 2, 6, 8, 8, 16, 28, 24, 12, 16, 28, 32, 48, 88

Offset: 0

N. J. A. Sloane, Jun 04 2009

nonn

			From _Omar E. Pol_, Jun 09 2009: (Start)
Triangle begins:
  1;
  2,2;
  2,6,8,4;
  2,6,8,8,16,28,24,8;
  2,6,8,8,16,28,24,12,16,28,32,48,88,104,64,16;
  2,6,8,8,16,28,24,12,16,28,32,48,88,104,64,20,16,28,32,48,88,104,72,56,88,...
(End)

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

For generating functions of the form Product_{k>=c} (1 + a*x^(2^k-1) + b*x^2^k) for the following values of (a,b,c) see: (1,1,0) A160573, (1,1,1) A151552, (1,1,2) A151692, (2,1,0) A151685, (2,1,1) A151691, (1,2,0) A151688 and A152980, (1,2,1) A151550, (2,2,0) A151693, (2,2,1) A151694.

Cf. A000079. - Omar E. Pol, Jun 09 2009

CoefficientList[Series[Product[1+2x^(2^k-1)+2x^2^k,{k,10}],{x,0,80}],x] (* Harvey P. Dale, Oct 07 2020 *)

A160571 G.f.: Product_{n>=1} (1 + x^n + x^(n+1)).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 15, 21, 28, 38, 52, 70, 92, 119, 154, 200, 258, 329, 416, 523, 655, 819, 1022, 1269, 1566, 1924, 2357, 2879, 3507, 4263, 5170, 6250, 7530, 9048, 10849, 12980, 15496, 18466, 21967, 26079, 30894, 36526, 43109, 50792, 59743, 70160

Offset: 0

Paul D. Hanna, May 20 2009, May 21 2009, Jul 17 2011

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 10*x^6 + 15*x^7 + ...
G.f.: A(x) = (1+x*(1+x))*(1+x^2*(1+x))*(1+x^3*(1+x))*(1+x^4*(1+x))*...
G.f.: A(x) = (1+x*(1+x)) + x^2*(1+x)*(1 + x^3*(1+x))*(1+x*(1+x))/(1-x) + x^7*(1+x)^2*(1 + x^5*(1+x))*(1+x*(1+x))*(1+x^2*(1+x))/((1-x)*(1-x^2)) + x^15*(1+x)^3*(1 + x^7*(1+x))*(1+x*(1+x))*(1+x^2*(1+x))*(1+x^3*(1+x))/((1-x)*(1-x^2)*(1-x^3)) + ...
G.f.: A(x) = 1 + x*(1+x)/(1-x) + x^3*(1+x)^2/((1-x)*(1-x^2)) + x^6*(1+x)^3/((1-x)*(1-x^2)*(1-x^3)) + x^10*(1+x)^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) + ...

Robert Israel, Table of n, a(n) for n = 0..10000
MathOverflow, Being even or odd in the product expansion, 2017.

Cf. A151552, A227681.

N:= 100: # for a(0)..a(N)
P:= mul(1+x^n+x^(n+1),n=1..N):
S:= series(P,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Sep 04 2018

With[{nn=50},CoefficientList[Series[Product[1+x^n+x^(n+1),{n,1,nn}],{x,0,nn}],x]] (* Harvey P. Dale, Dec 29 2015 *)

a(n)=polcoeff(prod(k=1,n,1+x^k*(1+x) +x*O(x^n)),n)

{a(n)=local(A=1+x); A=sum(m=0, n, x^(m*(3*m+1)/2)*(1+x)^m*(1 + x^(2*m+1)*A)*prod(k=1, m, (1+A*x^k)/(1-x^k+x*O(x^n)))); polcoeff(A, n)}

{a(n)=local(A=1+x); A=sum(m=0, n, x^(m*(m+1)/2)*(1+x)^m/prod(k=1, m, 1-x^k +x*O(x^n))); polcoeff(A, n)}

G.f.: A(x) = Sum_{n>=0} x^(n*(3*n+1)/2)*(1+x)^n*(1 + x^(2*n+1)*(1+x)) * Product_{k=1..n} (1 + x^k*(1+x))/(1-x^k) due to Sylvester's identity.
G.f.: A(x) = Sum_{n>=0} x^(n*(n+1)/2)*(1+x)^n / Product_{k=1..n} (1-x^k).
G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (-1)^(d+1)*(1 + x)^d/d). - Ilya Gutkovskiy, Apr 18 2019
a(n) ~ c * exp(r*sqrt(n)) / n^(3/4), where r = 2*sqrt(-polylog(2,-2)) = 2.397287105779... and c = (-polylog(2,-2))^(1/4) / (6*sqrt(Pi)) = 0.10294821957... - Vaclav Kotesovec, Oct 24 2020, updated Jun 25 2021

A151553 G.f.: (1 + x) * Product_{n>=1} (1 + x^(2^n-1) + x^(2^n)).

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 3, 2, 3, 4, 4, 5, 7, 7, 4, 2, 3, 4, 4, 5, 7, 7, 5, 5, 7, 8, 9, 12, 14, 11, 5, 2, 3, 4, 4, 5, 7, 7, 5, 5, 7, 8, 9, 12, 14, 11, 6, 5, 7, 8, 9, 12, 14, 12, 10, 12, 15, 17, 21, 26, 25, 16, 6, 2, 3, 4, 4, 5, 7, 7, 5, 5, 7, 8, 9, 12, 14, 11, 6, 5, 7, 8, 9, 12, 14, 12, 10, 12, 15, 17, 21, 26

Offset: 0

N. J. A. Sloane, May 20 2009

			If formatted as a triangle:
.1,
.2,
.2,2,
.3,4,3,2,
.3,4,4,5,7,7,4,2,
.3,4,4,5,7,7,5,5,7,8,9,12,14,11,5,2,
.3,4,4,5,7,7,5,5,7,8,9,12,14,11,6,5,7,8,9,12,14,12,10,12,15,17,21,26,25,16,6,2,
.3,4,4,5,7,7,5,5,7,8,9,12,14,11,6,5,7,8,9,12,14,12,10,12,15,17,21,26,25,16,7
... 5,7,8,9,12,14,12,10,12,15,17,21,26,25,17,11,12,15,17,21,26,26,22,22,27,32,38,47,51,41,22,7,2,
.3,4,4,5,7,7,4,2, ...

N. J. A. Sloane, Table of n, a(n) for n = 0..16383
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS.

Cf. A139250, A151550, A151551, A151552.

CoefficientList[Series[(1+x)Product[1+x^(2^n-1)+x^2^n,{n,10}],{x,0,100}],x] (* Harvey P. Dale, Jul 13 2019 *)
a[n_] := Sum[If[OddQ[n + k], Binomial[DigitCount[n + k, 2, 1], k], 0], {k, 0, 2*Floor[Log2[n + 1]] + 1}]; Array[a, 92, 0] (* Amiram Eldar, Jul 29 2023 *)

Recurrence: a(0)=1, a(1) = a(2) = 2; a(2^m-1)=2 for m >= 2; a(2^m) = 3 for m >= 2; a(2^m-2) = m for m >= 3; otherwise, for m >= 5, if m=2^i+j (0 <= j < 2^i - 1), a(m) = a(j) + a(j+1).

a(n) = Sum_{k>=0, n+k odd} binomial(A000120(n+k),k); the sum may be restricted further to k <= 2*A000523(n+1)+1 [based on Hagen von Eitzen's formula for A151552]. [corrected by Amiram Eldar, Jul 29 2023]

A151551 G.f.: (1 + 3x) * Product_{n>=1} (1 + x^(2^n-1) + 2*x^(2^n)).

Original entry on oeis.org

1, 4, 5, 7, 6, 13, 16, 13, 6, 13, 17, 20, 25, 42, 44, 25, 6, 13, 17, 20, 25, 42, 45, 32, 25, 43, 54, 65, 92, 128, 112, 49, 6, 13, 17, 20, 25, 42, 45, 32, 25, 43, 54, 65, 92, 128, 113, 56, 25, 43, 54, 65, 92, 129, 122, 89, 93, 140, 173, 222, 312, 368, 272, 97, 6, 13, 17, 20, 25, 42

Offset: 0

N. J. A. Sloane, May 19 2009

nonn

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A151550, A151552.

A151555 G.f.: (1 + 2x) * Product_{n>=1} (1 + x^(2^n-1) + 2*x^(2^n)).

Original entry on oeis.org

1, 3, 4, 5, 5, 10, 12, 9, 5, 10, 13, 15, 20, 32, 32, 17, 5, 10, 13, 15, 20, 32, 33, 23, 20, 33, 41, 50, 72, 96, 80, 33, 5, 10, 13, 15, 20, 32, 33, 23, 20, 33, 41, 50, 72, 96, 81, 39, 20, 33, 41, 50, 72, 97, 89, 66, 73, 107, 132, 172, 240, 272, 192, 65, 5, 10, 13, 15, 20, 32, 33, 23

Offset: 0

N. J. A. Sloane, May 20 2009

nonn

From Gary W. Adamson, May 25 2009: (Start)
Convolved with A078008 signed (A151575) [1, 0, 2, -2, 6, -10, 22, -42, 86, -170, ...]
equals the toothpick sequence A153006: (1, 3, 6, 9, 13, 20, 28, ...). (End)
If A151550 is written as a triangle then the rows converge to this sequence. - N. J. A. Sloane, Jun 16 2009

			From _Omar E. Pol_, Jun 19 2009: (Start)
May be written as a triangle:
1;
3;
4,5;
5,10,12,9;
5,10,13,15,20,32,32,17;
5,10,13,15,20,32,33,23,20,33,41,50,72,96,80,33;
5,10,13,15,20,32,33,23,20,33,41,50,72,96,81,39,20,33,41,50,72,97,89,66,73,...
(End)

N. J. A. Sloane, Table of n, a(n) for n = 0..16384
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A151551, A151552, A151553, A151554, A151550, A152980, A153006.

Cf. A078008. - Gary W. Adamson, May 25 2009

A151554 G.f.: (1 + 2x) * Product_{n>=1} (1 + x^(2^n-1) + x^(2^n)).

Original entry on oeis.org

1, 3, 3, 3, 4, 6, 5, 3, 4, 6, 6, 7, 10, 11, 7, 3, 4, 6, 6, 7, 10, 11, 8, 7, 10, 12, 13, 17, 21, 18, 9, 3, 4, 6, 6, 7, 10, 11, 8, 7, 10, 12, 13, 17, 21, 18, 10, 7, 10, 12, 13, 17, 21, 19, 15, 17, 22, 25, 30, 38, 39, 27, 11, 3, 4, 6, 6, 7, 10, 11, 8, 7, 10, 12, 13, 17, 21, 18, 10, 7, 10, 12, 13

Offset: 0

N. J. A. Sloane, May 20 2009

nonn

N. J. A. Sloane, Table of n, a(n) for n = 0..16383
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A151550, A151551, A151552, A151553.

A151684 Let c(n) = x^(2^n-1)(1-x^(2^n)), g(n) = 1 + x^(2^n-1) + x^(2^n), h(n) = Product_{i=1..n} g(i); then use g.f. (1+2x) - Sum_{n>=1} c(n)/h(n).

Original entry on oeis.org

1, 1, 1, 0, -1, 1, 1, -1, 0, 0, -2, 1, 2, 0, 2, -1, -5, -1, 1, 3, 7, 1, -8, -8, -6, 3, 18, 16, 0, -17, -31, -21, 19, 51, 47, 3, -70, -106, -48, 71, 170, 156, -18, -243, -318, -132, 253, 564, 455, -130, -819, -1024, -341, 952, 1849, 1355, -606, -2789, -3199, -727, 3410, 5979, 3932, -2678, -9408, -9926, -1281, 12047

Offset: 0

N. J. A. Sloane, Jun 01 2009

sign

This g.f. multiplied by h(k) for k large (cf. A151552) gives the g.f. for (A160573 prefixed by an initial 1).

Cf. A151676, A160573, A151552.

cp's OEIS Frontend

A151694 G.f.: Product_{k>=1} (1 + 2*x^(2^k-1) + 2*x^(2^k)).

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A160571 G.f.: Product_{n>=1} (1 + x^n + x^(n+1)).

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A151553 G.f.: (1 + x) * Product_{n>=1} (1 + x^(2^n-1) + x^(2^n)).

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A151551 G.f.: (1 + 3x) * Product_{n>=1} (1 + x^(2^n-1) + 2*x^(2^n)).

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A151555 G.f.: (1 + 2x) * Product_{n>=1} (1 + x^(2^n-1) + 2*x^(2^n)).

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A151554 G.f.: (1 + 2x) * Product_{n>=1} (1 + x^(2^n-1) + x^(2^n)).

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A151684 Let c(n) = x^(2^n-1)*(1-x^(2^n)), g(n) = 1 + x^(2^n-1) + x^(2^n), h(n) = Product_{i=1..n} g(i); then use g.f. (1+2*x) - Sum_{n>=1} c(n)/h(n).

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A151694 G.f.: Product_{k>=1} (1 + 2x^(2^k-1) + 2x^(2^k)).

A151684 Let c(n) = x^(2^n-1)(1-x^(2^n)), g(n) = 1 + x^(2^n-1) + x^(2^n), h(n) = Product_{i=1..n} g(i); then use g.f. (1+2x) - Sum_{n>=1} c(n)/h(n).