A151688 - OEIS

%I #18 Oct 04 2024 20:51:17

%S 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 46,56,70,104,128,96,34,8,14,18,20,30,44,42,28,30,46,56,70,104,128,98,

%U 44,30,46,56,70,104,130,112,86,106,148,182,244,336,352,224,66,8,14,18,20,30,44

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

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

%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.]

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

%e If written as a triangle, begins:

%e   2;

%e   4;

%e   6, 6;

%e   8, 14, 16, 10;

%e   8, 14, 18, 20, 30, 44, 40, 18;

%e   8, 14, 18, 20, 30, 44, 42, 28, 30, 46, 56, 70, 104, 128, 96, 34;

%e   ...

%Y Equals 2*A152980 = A147646/2.

%Y Equals limit of rows of triangle in A152968.

%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 Cf. A151550, A139251, A139250.

%K nonn,tabf

%O 0,1

%A _Omar E. Pol_, May 02 2009

%E Edited by _N. J. A. Sloane_, Jun 03 2009, Jul 14 2009