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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). 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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<k<l.

LINKS

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

Wikipedia, Fibonacci number

Wikipedia, Gray code

EXAMPLE

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...

PROG

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

CROSSREFS

Cf. A000045, A003188, A265385, A265386.

Sequence in context: A129784 A125050 A056186 * A267157 A054135 A171858

Adjacent sequences:  A265384 A265385 A265386 * A265388 A265389 A265390

KEYWORD

nonn

AUTHOR

Stanislav Sykora, Dec 07 2015

STATUS

approved

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Last modified June 28 16:56 EDT 2017. Contains 288839 sequences.