A279586 Coefficients in the expansion of ([r^2] + [2r^2]x + [3r^2]x^2 + ...)/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = golden ratio = (1 + sqrt(5))/2.

2, -1, 2, -4, 7, -12, 21, -36, 61, -104, 177, -300, 508, -860, 1455, -2460, 4158, -7027, 11873, -20058, 33884, -57237, 96679, -163296, 275811, -465843, 786796, -1328867, 2244390, -3790645, 6402161, -10812825, 18262114, -30843412, 52092289, -87980053

Offset: 0

Author

Clark Kimberling, Dec 15 2016

Keywords

• sign
• easy

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000201 (denominator coefficients, the lower Wythoff sequence), A001950 (numerator coefficients, the upper Wythoff sequence).

Programs

Mathematica

z = 30; r = GoldenRatio;
f[x_] := f[x] = Sum[Floor[r*(k + 1)] x^k, {k, 0, z}];
g[x_] := g[x] = Sum[Floor[(r^2)*(k + 1)] x^k, {k, 0, z}]; f[x]
CoefficientList[Series[g[x]/f[x], {x, 0, 2*z}], x]

PARI

r = (1 + sqrt(5))/2;
f(x) = sum(k=0, 36, floor(r*(k + 1))*x^k);
g(x) = sum(k=0, 36, floor((r^2)*(k + 1))*x^k);
x = 'x + O('x^36); Vec(g(x)/f(x)) \\ Indranil Ghosh, Apr 18 2017

Formula

G.f.: ([r^2] + [2r^2]x + [3r^2]x^2 + ...)/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = golden ratio - (1 + sqrt(5))/2.

G.f.: 1 + 1/(1 + x/(1 + x/(1 + x^2/(1 + x^3/(1 + x^5/(1 + x^8/(1 + ... + x^Fibonacci(k)/(1 + ... )))))))), a continued fraction (conjecture). - Ilya Gutkovskiy, Apr 18 2017

From Vaclav Kotesovec, Nov 17 2017: (Start)

a(n) ~ (-1)^n * c * d^n, where

d = 1.6889241107691652066863596437198336089614626466166721916456635666408929438...

c = 0.9506728401953553339263330409185586796133637973131016737129828085573272295...

(End)