A005908 a(n) = floor( phi*a(n-1) ) + floor( phi*a(n-2) ), where phi is the golden ratio.

0, 1, 1, 2, 4, 9, 20, 46, 106, 245, 567, 1313, 3041, 7044, 16317, 37798, 87559, 202831, 469860, 1088436, 2521375, 5840796, 13530276, 31343052, 72606569, 168194019, 389623535, 902567761, 2090809436, 4843386045

Offset

0,4

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Formula

Limit_{n -> infinity} a(n)/a(n-1) = phi/C = 2.31651242917313... where phi is the golden ratio and C the positive root of x^4 + x^3 - 2*x^2 - 2*x - 1 (C = 0.6984784404...). - Benoit Cloitre, Aug 13 2002

Conjectures from Chai Wah Wu, Jun 13 2020: (Start)

a(n) = 2*a(n-1) + a(n-2) - a(n-4) - a(n-5) for n > 4.

G.f.: x*(-x^3 - x^2 - x + 1)/(x^5 + x^4 - x^2 - 2*x + 1). (End)

Mathematica

nxt[{a_, b_}]:={b, Floor[GoldenRatio*a]+Floor[GoldenRatio*b]}; Transpose[ NestList[ nxt, {0, 1}, 30]][[1]] (* Harvey P. Dale, Apr 24 2016 *)

Crossrefs

Cf. A001622 (golden ratio).

Sequence in context: A191653 A191827 A000968 * A206119 A085748 A317097

Adjacent sequences: A005905 A005906 A005907 * A005909 A005910 A005911

Keyword

nonn

Author

N. J. A. Sloane

Status

approved