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%I #18 Jul 29 2023 03:13:19
%S 1,2,2,2,3,4,3,2,3,4,4,5,7,7,4,2,3,4,4,5,7,7,5,5,7,8,9,12,14,11,5,2,3,
%T 4,4,5,7,7,5,5,7,8,9,12,14,11,6,5,7,8,9,12,14,12,10,12,15,17,21,26,25,
%U 16,6,2,3,4,4,5,7,7,5,5,7,8,9,12,14,11,6,5,7,8,9,12,14,12,10,12,15,17,21,26
%N G.f.: (1 + x) * Product_{n>=1} (1 + x^(2^n-1) + x^(2^n)).
%H N. J. A. Sloane, <a href="/A151553/b151553.txt">Table of n, a(n) for n = 0..16383</a>
%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 Recurrence: a(0)=1, a(1) = a(2) = 2; a(2^m-1)=2 for m >= 2; a(2^m) = 3 for m >= 2; a(2^m-2) = m for m >= 3; otherwise, for m >= 5, if m=2^i+j (0 <= j < 2^i - 1), a(m) = a(j) + a(j+1).
%F a(n) = Sum_{k>=0, n+k odd} binomial(A000120(n+k),k); the sum may be restricted further to k <= 2*A000523(n+1)+1 [based on Hagen von Eitzen's formula for A151552]. [corrected by _Amiram Eldar_, Jul 29 2023]
%e If formatted as a triangle:
%e .1,
%e .2,
%e .2,2,
%e .3,4,3,2,
%e .3,4,4,5,7,7,4,2,
%e .3,4,4,5,7,7,5,5,7,8,9,12,14,11,5,2,
%e .3,4,4,5,7,7,5,5,7,8,9,12,14,11,6,5,7,8,9,12,14,12,10,12,15,17,21,26,25,16,6,2,
%e .3,4,4,5,7,7,5,5,7,8,9,12,14,11,6,5,7,8,9,12,14,12,10,12,15,17,21,26,25,16,7
%e ... 5,7,8,9,12,14,12,10,12,15,17,21,26,25,17,11,12,15,17,21,26,26,22,22,27,32,38,47,51,41,22,7,2,
%e .3,4,4,5,7,7,4,2, ...
%t CoefficientList[Series[(1+x)Product[1+x^(2^n-1)+x^2^n,{n,10}],{x,0,100}],x] (* _Harvey P. Dale_, Jul 13 2019 *)
%t a[n_] := Sum[If[OddQ[n + k], Binomial[DigitCount[n + k, 2, 1], k], 0], {k, 0, 2*Floor[Log2[n + 1]] + 1}]; Array[a, 92, 0] (* _Amiram Eldar_, Jul 29 2023 *)
%Y Cf. A139250, A151550, A151551, A151552.
%K nonn,tabf
%O 0,2
%A _N. J. A. Sloane_, May 20 2009