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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).
3
1, 1, 3, 2, 7, 4, 15, 9, 31, 19, 63, 39, 126, 79, 253, 158, 510, 315, 1012, 622, 2004, 1116, 4072, 2505, 8173, 5100, 16175, 10171, 32657, 20192, 64797, 39858, 128257, 71450, 260628, 160367, 523085, 326498, 1035105, 651126, 2090065, 1292517, 4146840
OFFSET
1,3
COMMENTS
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.
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.
LINKS
Wikipedia, Fibonacci number
Wikipedia, Gray code
EXAMPLE
r(10) = 1.417436..., r(1000) = 1.414393...
s(100) = 0.629..., s(101) = 3.210..., s(102) = 0.618...
s(10000) = 0.631..., s(10001) = 3.183..., s(10002) = 0.608...
PROG
(PARI) gray(m)=bitxor(m, m>>1);
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Stanislav Sykora, Dec 07 2015
STATUS
approved