A151550 Expansion of g.f. Product_{n >= 1} (1 + x^(2^n-1) + 2*x^(2^n)).

1, 1, 2, 1, 3, 4, 4, 1, 3, 4, 5, 5, 10, 12, 8, 1, 3, 4, 5, 5, 10, 12, 9, 5, 10, 13, 15, 20, 32, 32, 16, 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, 32, 1, 3, 4, 5, 5, 10, 12, 9, 5, 10, 13, 15, 20, 32, 32, 17, 5, 10, 13

Offset: 0

Author

N. J. A. Sloane, May 19 2009, Jun 17 2009

Keywords

• nonn

Comments

When convolved with [1, 2, 2, 2, ...] gives the toothpick sequence A153006: (1, 3, 6, 9, ...). - Gary W. Adamson, May 25 2009

This sequence and the Adamson's comment both are mentioned in the Applegate-Pol-Sloane article, see chapter 8 "generating functions". - Omar E. Pol, Sep 20 2011

Examples

			From _Omar E. Pol_, Jun 09 2009, edited by _N. J. A. Sloane_, Jun 17 2009:
May be written as a triangle:
  0;
  1;
  1,2;
  1,3,4,4;
  1,3,4,5,5,10,12,8;
  1,3,4,5,5,10,12,9,5,10,13,15,20,32,32,16;
  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,...
The rows of the triangle converge to A151555.
		

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

• 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.], which is also available at arXiv:1004.3036v2, [math.CO], 2010.
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

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.

Cf. A139250, A151551, A151552, A151553, A151554, A151555, A152980, A153006, A151688.

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

Programs

Mathematica

terms = 100;
CoefficientList[Product[(1+x^(2^n-1) + 2 x^(2^n)), {n, 1, Log[2, terms] // Ceiling}] + O[x]^terms, x] (* Jean-François Alcover, Aug 05 2018 *)

Formula

To get a nice recurrence, change the offset to 0 and multiply the g.f. by x as in the triangle in the example lines. Then we have: a(0)=0; a(2^i)=1; a(2^i-1)=2^(i-1) for i >= 1; otherwise write n = 2^i+j with 1 <= j <= 2^i-2, then a(n) = a(2^i+j) = 2*a(j) + a(j+1).