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A265386 Sequence defined by a(1)=a(2)=1 and a(n) = gray(gray(a(n-1)) + gray(a(n-2))), with gray(m) = A003188(m).

%I A265386 #16 Jan 16 2016 18:08:45
%S A265386 1,1,3,2,7,4,15,9,31,19,63,39,126,79,253,158,510,315,1012,622,2004,
%T A265386 1116,4072,2505,8173,5100,16175,10171,32657,20192,64797,39858,128257,
%U A265386 71450,260628,160367,523085,326498,1035105,651126,2090065,1292517,4146840
%N A265386 Sequence defined by a(1)=a(2)=1 and a(n) = gray(gray(a(n-1)) + gray(a(n-2))), with gray(m) = A003188(m).
%C A265386 This recurrence is reminiscent of Fibonacci's, except that in each step the arguments as well as the result are passed through the binary-reflected Gray code mapping, which introduces a degree of pseudo-randomness.
%C A265386 Conjecture: the mean growth rate r(n) = (a(2n)/a(n))^(1/n) appears to converge to sqrt(2), with the consecutive-terms ratio s(n) = a(n)/a(n-1) exhibiting large and persistent fluctuations around the mean value.
%H A265386 Stanislav Sykora, <a href="/A265386/b265386.txt">Table of n, a(n) for n = 1..1000</a>
%H A265386 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fibonacci_number">Fibonacci number</a>
%H A265386 Wikipedia, <a href="https://en.wikipedia.org/wiki/Gray_code">Gray code</a>
%e A265386 r(10) = 1.417436..., r(1000) = 1.414393...
%e A265386 s(100) = 0.629..., s(101) = 3.210..., s(102) = 0.618...
%e A265386 s(10000) = 0.631..., s(10001) = 3.183..., s(10002) = 0.608...
%o A265386 (PARI) gray(m)=bitxor(m,m>>1);
%o A265386 a=vector(1000);a[1]=1;a[2]=1;for(n=3,#a,a[n]=gray(gray(a[n-1])+gray(a[n-2])));a
%Y A265386 Cf. A000045, A003188, A265385, A265387.
%K A265386 nonn
%O A265386 1,3
%A A265386 _Stanislav Sykora_, Dec 07 2015