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A151702 a(0)=1, a(1)=0; a(2^i + j) = a(j) + a(j+1) for 0 <= j < 2^i.

%I A151702 #12 Feb 24 2021 02:48:18
%S A151702 1,0,1,1,1,1,2,2,1,1,2,2,2,3,4,3,1,1,2,2,2,3,4,3,2,3,4,4,5,7,7,4,1,1,
%T A151702 2,2,2,3,4,3,2,3,4,4,5,7,7,4,2,3,4,4,5,7,7,5,5,7,8,9,12,14,11,5,1,1,2,
%U A151702 2,2,3,4,3,2,3,4,4,5,7,7,4,2,3,4,4,5,7,7,5,5,7,8,9,12,14,11,5,2,3,4,4,5,7,7
%N A151702 a(0)=1, a(1)=0; a(2^i + j) = a(j) + a(j+1) for 0 <= j < 2^i.
%H A151702 Ivan Neretin, <a href="/A151702/b151702.txt">Table of n, a(n) for n = 0..8191</a>
%H A151702 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 A151702 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>
%p A151702 f:=proc(r,s,a,b) local s1,n,i,j;
%p A151702 s1:=array(0..120);
%p A151702 s1[0]:=r; s1[1]:=s;
%p A151702 for n from 2 to 120 do i:=floor(log(n)/log(2));
%p A151702 j:=n-2^i; s1[n]:=a*s1[j]+b*s1[j+1]; od:
%p A151702 [seq(s1[n],n=0..120)];
%p A151702 end;
%p A151702 f(1,0,1,1);
%t A151702 a = {1, 0}; Do[AppendTo[a, a[[j]] + a[[j + 1]]], {i, 6}, {j, 2^i}]; a (* _Ivan Neretin_, Jun 28 2017 *)
%Y A151702 For the recurrence a(2^i+j) = C*a(j) + D*a(j+1), a(0) = A, a(1) = B for following values of (A B C D) see: (0 1 1 1) A118977, (1 0 1 1) A151702, (1 1 1 1) A151570, (1 2 1 1) A151571, (0 1 1 2) A151572, (1 0 1 2) A151703, (1 1 1 2) A151573, (1 2 1 2) A151574, (0 1 2 1) A160552, (1 0 2 1) A151704, (1 1 2 1) A151568, (1 2 2 1) A151569, (0 1 2 2) A151705, (1 0 2 2) A151706, (1 1 2 2) A151707, (1 2 2 2) A151708.
%Y A151702 If first two terms are dropped, same as A151552.
%K A151702 nonn
%O A151702 0,7
%A A151702 _N. J. A. Sloane_, Jun 06 2009