A141528 Expansion of x/(1 + x + 41*x^2).

0, -1, 1, 40, -81, -1559, 4880, 59039, -259119, -2161480, 12785359, 75835321, -600035040, -2509213121, 27110649761, 75767088200, -1187303728401, -1919146887799, 50598599752240, 28086422647519, -2102629012489359, 951085683941080, 85256703828122639

Offset

1,4

Links

G. C. Greubel, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (-1,-41).

Formula

a(n) = (-1)^(n-1)*(p^n - q^n)/(p-q), where p = (1 + sqrt(163)*i)/2, q = (1 - sqrt(163)*i)/2.

G.f.: x/(1 + x + 41*x^2). - Roger L. Bagula, Apr 18 2010

a(n) = -a(n-1) -41*a(n-2), with a(0) = 0, a(1) = -1. - G. C. Greubel, Mar 29 2021

Mathematica

p:= (1 +Sqrt[163]*I)/2; q:= (1 -Sqrt[163]*I)/2; f[n_]:= (-1)^(n-1)*(p^n -q^n)/(p-q); Table[Simplify[f[n]], {n, 0, 30}] (* modified by G. C. Greubel, Mar 29 2021 *)
CoefficientList[Series[x/(1+x+41*x^2), {x, 0, 30}], x] (* Roger L. Bagula, Apr 18 2010; modified by G. C. Greubel, Mar 29 2021 *)
LinearRecurrence[{-1, -41}, {0, -1}, 30] (* G. C. Greubel, Mar 29 2021 *)

Prog

(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( x/(1+x+41*x^2) )); // G. C. Greubel, Mar 29 2021
(Sage)
def A141528_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( x/(1+x+41*x^2) ).list()
a=A141528_list(31); a[1:] # G. C. Greubel, Mar 29 2021

Crossrefs

Cf. A005846, A141527.

Sequence in context: A181458 A069070 A174052 * A160282 A243803 A203855

Adjacent sequences: A141525 A141526 A141527 * A141529 A141530 A141531

Keyword

sign

Author

Roger L. Bagula, Aug 11 2008

Extensions

Edited by G. C. Greubel, Mar 29 2021

Status

approved