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

%I A151688 #22 Sep 02 2025 21:57:36
%S A151688 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,
%T A151688 46,56,70,104,128,96,34,8,14,18,20,30,44,42,28,30,46,56,70,104,128,98,
%U A151688 44,30,46,56,70,104,130,112,86,106,148,182,244,336,352,224,66,8,14,18,20,30,44
%N A151688 G.f.: Product_{n>=0} (1 + x^(2^n-1) + 2*x^(2^n)).
%C A151688 This is essentially the same g.f. as A151550 but with the n=0 term included.
%H A151688 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.]
%H A151688 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 A151688 a(n) = Sum_{k>=0} 2^(wt(n+k)-k)*binomial(wt(n+k),k).
%e A151688 If written as a triangle, begins:
%e A151688   2;
%e A151688   4;
%e A151688   6, 6;
%e A151688   8, 14, 16, 10;
%e A151688   8, 14, 18, 20, 30, 44, 40, 18;
%e A151688   8, 14, 18, 20, 30, 44, 42, 28, 30, 46, 56, 70, 104, 128, 96, 34;
%e A151688   ...
%t A151688 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 *)
%Y A151688 Equals 2*A152980 = A147646/2.
%Y A151688 Equals limit of rows of triangle in A152968.
%Y A151688 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 A151688 Cf. A151550, A139251, A139250.
%K A151688 nonn,tabf,changed
%O A151688 0,1
%A A151688 _Omar E. Pol_, May 02 2009
%E A151688 Edited by _N. J. A. Sloane_, Jun 03 2009, Jul 14 2009