A292461 Expansion of (1 - x - x^2 + sqrt((1 - x - x^2)^2 - 4*x^3))/2 in powers of x.

1, -1, -1, -1, -1, -2, -4, -8, -17, -37, -82, -185, -423, -978, -2283, -5373, -12735, -30372, -72832, -175502, -424748, -1032004, -2516347, -6155441, -15101701, -37150472, -91618049, -226460893, -560954047, -1392251012, -3461824644, -8622571758, -21511212261

Offset

0,6

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

Convolution inverse of A292460.

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

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).

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

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

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

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

Mathematica

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 *)

Prog

(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
(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

Crossrefs

Cf. A004148, A203019, A292440, A292460.

Sequence in context: A024557 A199409 A025241 * A203019 A004148 A292460

Adjacent sequences: A292458 A292459 A292460 * A292462 A292463 A292464

Keyword

sign

Author

Seiichi Manyama, Sep 16 2017

Status

approved