A151688 G.f.: Product_{n>=0} (1 + x^(2^n-1) + 2*x^(2^n)).

2, 4, 6, 6, 8, 14, 16, 10, 8, 14, 18, 20, 30, 44, 40, 18, 8, 14, 18, 20, 30, 44, 42, 28, 30, 46, 56, 70, 104, 128, 96, 34, 8, 14, 18, 20, 30, 44, 42, 28, 30, 46, 56, 70, 104, 128, 98, 44, 30, 46, 56, 70, 104, 130, 112, 86, 106, 148, 182, 244, 336, 352, 224, 66, 8, 14, 18, 20, 30, 44

Offset: 0

Author

Omar E. Pol, May 02 2009

Keywords

• nonn
• tabf
• changed

Comments

This is essentially the same g.f. as A151550 but with the n=0 term included.

Examples

			If written as a triangle, begins:
  2;
  4;
  6, 6;
  8, 14, 16, 10;
  8, 14, 18, 20, 30, 44, 40, 18;
  8, 14, 18, 20, 30, 44, 42, 28, 30, 46, 56, 70, 104, 128, 96, 34;
  ...
		

Links

• 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

Crossrefs

Equals 2*A152980 = A147646/2.

Equals limit of rows of triangle in A152968.

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. A151550, A139251, A139250.

Programs

Mathematica

terms = 70; CoefficientList[Product[(1+x^(2^n-1) + 2 x^(2^n)), {n, 0, Log[2, terms] // Ceiling}] + O[x]^terms, x] (* Stefano Spezia, Sep 02 2025 *)

Formula

a(n) = Sum_{k>=0} 2^(wt(n+k)-k)*binomial(wt(n+k),k).

Extensions

Edited by N. J. A. Sloane, Jun 03 2009, Jul 14 2009