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).

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

Author

Stanislav Sykora, Dec 07 2015

Keywords

• nonn

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.

Examples

			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...
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..1000
• Wikipedia, Fibonacci number
• Wikipedia, Gray code

Crossrefs

Cf. A000045, A003188, A265385, A265387.

Programs

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