A265385 -id:A265385 - cp's OEIS frontend

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

Stanislav Sykora, Dec 07 2015

nonn

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.

			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...

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

Cf. A000045, A003188, A265385, A265387.

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

A265387 Sequence defined by a(1)=a(2)=1 and a(n) = gray(a(n-1)) + gray(a(n-2)), with gray(m) = A003188(m).

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 39, 78, 157, 316, 629, 1265, 2520, 5053, 10135, 20159, 40508, 80642, 161701, 324346, 645118, 1296264, 2580557, 5174455, 10379095, 20643816, 41480472, 82577840, 165582588, 332131050, 660602145, 1327375184, 2642491049, 5298643189

Offset: 1

Stanislav Sykora, Dec 07 2015

nonn

This recurrence is reminiscent of Fibonacci's, except that in each step the arguments 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 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

			r(10) = 2.000470476732..., r(1000) = 2.000000000203...
s(100) = 2.0058315..., s(101) = 1.9889791..., s(102) = 2.0093437...
s(10000) = 2.0058331..., s(10001) = 1.9889803..., s(10002) = 2.0093413...

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

Cf. A000045, A003188, A265385, A265386.

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])+gray(a[n-2]));a

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cp's OEIS Frontend

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).

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