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%I #26 Feb 24 2021 02:48:18
%S 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,
%T 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,
%U 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
%N G.f.: Product_{ k >= 0} (1 + x^(2^k-1) + x^(2^k)).
%C Sequence mentioned in the Applegate-Pol-Sloane article; see Section 9, "explicit formulas." - _Omar E. Pol_, Sep 20 2011
%D 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
%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, 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 <a href="http://arxiv.org/abs/1004.3036">arXiv:1004.3036v2</a>
%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%F a(n) = Sum_{i >= 0} binomial(A000120(n+i),i).
%F 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).
%F a(n) = 2*A151552(n) + A151552(n-1).
%e 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
%e From _Omar E. Pol_, Jun 09 2009: (Start)
%e Triangle begins:
%e 2;
%e 3;3;
%e 3,5,6,4;
%e 3,5,6,6,8,11,10,5;
%e 3,5,6,6,8,11,10,7,8,11,12,14,19,21,15,6;
%e 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,...
%e (End)
%p See A118977 for Maple code.
%t 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 *)
%Y 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
%Y Row sums of A151683. See A151687 for another version.
%Y Cf. A151552 (g.f. has one factor fewer).
%Y 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
%Y Cf. A139250, A139251. - _Omar E. Pol_, Sep 20 2011
%K nonn
%O 0,1
%A _Hagen von Eitzen_, May 20 2009
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