A160573 G.f.: Product_{k >= 0} (1 + x^(2^k-1) + x^(2^k)).

2, 3, 3, 3, 5, 6, 4, 3, 5, 6, 6, 8, 11, 10, 5, 3, 5, 6, 6, 8, 11, 10, 7, 8, 11, 12, 14, 19, 21, 15, 6, 3, 5, 6, 6, 8, 11, 10, 7, 8, 11, 12, 14, 19, 21, 15, 8, 8, 11, 12, 14, 19, 21, 17, 15, 19, 23, 26, 33, 40, 36, 21, 7, 3, 5, 6, 6, 8, 11, 10, 7, 8, 11, 12, 14, 19, 21, 15, 8, 8

Offset

0,1

Comments

Sequence mentioned in the Applegate-Pol-Sloane article; see Section 9, "explicit formulas." - Omar E. Pol, Sep 20 2011

References

D. 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

Links

Table of n, a(n) for n=0..79.

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.], which is also available at arXiv:1004.3036v2

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Formula

a(n) = Sum_{i >= 0} binomial(A000120(n+i),i).

For k >= 1, a(2^k-2) = k+1 and a(2^k-1) = 3; otherwise if n = 2^i + j, 0 <= j <= 2^i-3, a(n) = a(j) + a(j+1).

a(n) = 2*A151552(n) + A151552(n-1).

Example

a(5) = binomial(2,0) + binomial(2,1) + binomial(3,2) + binomial(1,3) + binomial(2,4) + binomial(2,5) + ... = 1 + 2 + 3 + 0 + 0 + 0 + ... = 6
From Omar E. Pol, Jun 09 2009: (Start)
Triangle begins:
2;
3;3;
3,5,6,4;
3,5,6,6,8,11,10,5;
3,5,6,6,8,11,10,7,8,11,12,14,19,21,15,6;
3,5,6,6,8,11,10,7,8,11,12,14,19,21,15,8,8,11,12,14,19,21,17,15,19,23,26,...
(End)

Maple

See A118977 for Maple code.

Mathematica

max = 80; Product[1 + x^(2^k - 1) + x^(2^k), {k, 0, Ceiling[Log[2, max]]}] + O[x]^max // CoefficientList[#, x]& (* Jean-François Alcover, Nov 10 2016 *)

Crossrefs

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.

Row sums of A151683. See A151687 for another version.

Cf. A151552 (g.f. has one factor fewer).

Limiting form of rows of A118977 when that sequence is written as a triangle and the initial 1 is omitted. - N. J. A. Sloane, Jun 01 2009

Cf. A139250, A139251. - Omar E. Pol, Sep 20 2011

Sequence in context: A338451 A145281 A151687 * A141418 A287771 A335107

Adjacent sequences: A160570 A160571 A160572 * A160574 A160575 A160576

Keyword

nonn

Author

Hagen von Eitzen, May 20 2009

Status

approved