A292461 - OEIS

%I #26 Sep 08 2022 08:46:19

%S 1,-1,-1,-1,-1,-2,-4,-8,-17,-37,-82,-185,-423,-978,-2283,-5373,-12735,

%T -30372,-72832,-175502,-424748,-1032004,-2516347,-6155441,-15101701,

%U -37150472,-91618049,-226460893,-560954047,-1392251012,-3461824644,-8622571758,-21511212261

%N Expansion of (1 - x - x^2 + sqrt((1 - x - x^2)^2 - 4*x^3))/2 in powers of x.

%H G. C. Greubel, <a href="/A292461/b292461.txt">Table of n, a(n) for n = 0..1000</a>

%F Convolution inverse of A292460.

%F Let f(x) = (1 - x - x^2 - sqrt((1 - x - x^2)^2 - 4*x^3))/(2*x^3).

%F G.f.: 1-x-x^2-x^3/(1-x-x^2-x^3/(1-x-x^2-x^3/(1-x-x^2-x^3/(1-x-x^2-x^3/(... (continued fraction).

%F G.f.: 1/f(x) = 1 - x - x^2 - x^3*f(x).

%F a(n) = -A292460(n-3) for n > 2.

%F a(n) ~ -5^(1/4) * phi^(2*n - 2) / (2 * sqrt(Pi) * n^(3/2)), where phi is the golden ratio (1+sqrt(5))/2. - _Vaclav Kotesovec_, Sep 17 2017, simplified Dec 06 2021

%F Conjecture D-finite with recurrence: n*a(n) +(-2*n+3)*a(n-1) +(-n+3)*a(n-2) +(-2*n+9)*a(n-3) +(n-6)*a(n-4)=0. - _R. J. Mathar_, Jan 24 2020

%t CoefficientList[Series[(1-x-x^2 +Sqrt[(1-x-x^2)^2 -4*x^3])/2, {x, 0, 50} ], x] (* _G. C. Greubel_, Aug 13 2018 *)

%o (PARI) x='x+O('x^50); Vec((1-x-x^2 +sqrt((1-x-x^2)^2 -4*x^3))/2) \\ _G. C. Greubel_, Aug 13 2018

%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x-x^2 +Sqrt((1-x-x^2)^2 -4*x^3))/2)); // _G. C. Greubel_, Aug 13 2018

%Y Cf. A004148, A203019, A292440, A292460.

%K sign

%O 0,6

%A _Seiichi Manyama_, Sep 16 2017