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A265387 Sequence defined by a(1)=a(2)=1 and a(n) = gray(a(n-1)) + gray(a(n-2)), with gray(m) = A003188(m).

%I A265387 #15 Jan 16 2016 17:47:49
%S A265387 1,1,2,4,9,19,39,78,157,316,629,1265,2520,5053,10135,20159,40508,
%T A265387 80642,161701,324346,645118,1296264,2580557,5174455,10379095,20643816,
%U A265387 41480472,82577840,165582588,332131050,660602145,1327375184,2642491049,5298643189
%N A265387 Sequence defined by a(1)=a(2)=1 and a(n) = gray(a(n-1)) + gray(a(n-2)), with gray(m) = A003188(m).
%C A265387 This recurrence is reminiscent of Fibonacci's, except that in each step the arguments are passed through the binary-reflected Gray code mapping, which introduces a degree of pseudo-randomness.
%C A265387 Conjecture: The mean growth rate r(n) = (a(2n)/a(n))^(1/n) appears to converge exactly to 2, with the consecutive-terms ratio s(n) = a(n)/a(n-1) exhibiting relatively small (~1%) but persistent fluctuations around the mean value. This contrasts what one might first expect, that sequence's growth rate were similar to that of the Fibonacci sequence, i.e., the golden ratio, since gray(m) just permutes every block of numbers ranging from 2^k to 2^l-1, for any 0<k<l.
%H A265387 Stanislav Sykora, <a href="/A265387/b265387.txt">Table of n, a(n) for n = 1..1000</a>
%H A265387 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fibonacci_number">Fibonacci number</a>
%H A265387 Wikipedia, <a href="https://en.wikipedia.org/wiki/Gray_code">Gray code</a>
%e A265387 r(10) = 2.000470476732..., r(1000) = 2.000000000203...
%e A265387 s(100) = 2.0058315..., s(101) = 1.9889791..., s(102) = 2.0093437...
%e A265387 s(10000) = 2.0058331..., s(10001) = 1.9889803..., s(10002) = 2.0093413...
%o A265387 (PARI) gray(m)=bitxor(m,m>>1);
%o A265387 a=vector(1000);a[1]=1;a[2]=1;for(n=3,#a,a[n]=gray(a[n-1])+gray(a[n-2]));a
%Y A265387 Cf. A000045, A003188, A265385, A265386.
%K A265387 nonn
%O A265387 1,3
%A A265387 _Stanislav Sykora_, Dec 07 2015