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A265385 Sequence defined by a(1)=a(2)=1 and a(n) = gray(a(n-1) + a(n-2)), with gray(m) = A003188(m).

%I A265385 #18 Jan 16 2016 18:01:31
%S A265385 1,1,3,6,13,26,52,105,211,418,847,1673,3380,6755,13404,27104,53538,
%T A265385 108163,216183,428935,867329,1713228,3461227,6917868,13725948,
%U A265385 27754524,54823316,110759272,221371778,439230367,888144817,1754346232,3544296957,7083888783
%N A265385 Sequence defined by a(1)=a(2)=1 and a(n) = gray(a(n-1) + a(n-2)), with gray(m) = A003188(m).
%C A265385 This recurrence is reminiscent of Fibonacci's, except that the result of each step is passed through the binary-reflected Gray code mapping, which introduces a degree of pseudo-randomness.
%C A265385 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 A265385 Stanislav Sykora, <a href="/A265385/b265385.txt">Table of n, a(n) for n = 1..1000</a>
%H A265385 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fibonacci_number">Fibonacci number</a>
%H A265385 Wikipedia, <a href="https://en.wikipedia.org/wiki/Gray_code">Gray code</a>
%e A265385 r(10) = 2.000421531046..., r(1000) = 1.999999999903...
%e A265385 s(100) = 1.9841292..., s(101) = 2.0220518..., s(102) = 1.9752921...
%e A265385 s(10000) = 1.9841299..., s(10001) = 2.0220478..., s(10002) = 1.9752929...
%o A265385 (PARI) gray(m)=bitxor(m,m>>1);
%o A265385 a=vector(1000);a[1]=1;a[2]=1;for(n=3,#a,a[n]=gray(a[n-1]+a[n-2]));a
%Y A265385 Cf. A000045, A003188, A265386, A265387.
%K A265385 nonn
%O A265385 1,3
%A A265385 _Stanislav Sykora_, Dec 07 2015