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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

%I

%S 2,-1,2,-4,7,-12,21,-36,61,-104,177,-300,508,-860,1455,-2460,4158,

%T -7027,11873,-20058,33884,-57237,96679,-163296,275811,-465843,786796,

%U -1328867,2244390,-3790645,6402161,-10812825,18262114,-30843412,52092289,-87980053

%N 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.

%H Clark Kimberling, <a href="/A279586/b279586.txt">Table of n, a(n) for n = 0..1000</a>

%F 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.

%F 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

%F From _Vaclav Kotesovec_, Nov 17 2017: (Start)

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

%F d = 1.6889241107691652066863596437198336089614626466166721916456635666408929438...

%F c = 0.9506728401953553339263330409185586796133637973131016737129828085573272295...

%F (End)

%t z = 30; r = GoldenRatio;

%t f[x_] := f[x] = Sum[Floor[r*(k + 1)] x^k, {k, 0, z}];

%t g[x_] := g[x] = Sum[Floor[(r^2)*(k + 1)] x^k, {k, 0, z}]; f[x]

%t CoefficientList[Series[g[x]/f[x], {x, 0, 2*z}], x]

%o (PARI) r = (1 + sqrt(5))/2;

%o f(x) = sum(k=0, 36, floor(r*(k + 1))*x^k);

%o g(x) = sum(k=0, 36, floor((r^2)*(k + 1))*x^k);

%o x = 'x + O('x^36); Vec(g(x)/f(x)) \\ _Indranil Ghosh_, Apr 18 2017

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

%K sign,easy

%O 0,1

%A _Clark Kimberling_, Dec 15 2016

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Last modified February 21 14:40 EST 2018. Contains 299414 sequences. (Running on oeis4.)