1,3

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.

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.

Stanislav Sykora, Table of n, a(n) for n = 1..1000

Wikipedia, Fibonacci number

Wikipedia, Gray code

r(10) = 2.000421531046..., r(1000) = 1.999999999903...

s(100) = 1.9841292..., s(101) = 2.0220518..., s(102) = 1.9752921...

s(10000) = 1.9841299..., s(10001) = 2.0220478..., s(10002) = 1.9752929...

(PARI) gray(m)=bitxor(m, m>>1);

a=vector(1000); a[1]=1; a[2]=1; for(n=3, #a, a[n]=gray(a[n-1]+a[n-2])); a

Cf. A000045, A003188, A265386, A265387.

Sequence in context: A079941 A255125 A267367 * A019300 A072762 A081254

Adjacent sequences: A265382 A265383 A265384 * A265386 A265387 A265388

Stanislav Sykora, Dec 07 2015

approved