A110310 Expansion of (1-x+x^2)/((x^2+x+1)*(x^2+5*x+1)).

1, -7, 36, -173, 827, -3960, 18973, -90907, 435564, -2086913, 9998999, -47908080, 229541401, -1099798927, 5269453236, -25247467253, 120967883027, -579591947880, 2776991856373, -13305367333987, 63749844813564, -305443856733833, 1463469438855599, -7011903337544160

Offset

0,2

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (-6,-7,-6,-1).

Formula

a(n+2) = - 5*a(n+1) - a(n) - (-1)^n*A109265(n+3).

a(n) = -6*a(n-1) - 7*a(n-2) - 6*a(n-3) - a(n-4) for n>3. - Colin Barker, Apr 30 2019

a(n) = (1/2)*(3*ChevyshevU(n, -5/2) - ChebyshevU(n, -1/2)). - G. C. Greubel, Jan 02 2023

Maple

seriestolist(series((1-x+x^2)/((x^2+x+1)*(x^2+5*x+1)), x=0, 25));

Mathematica

LinearRecurrence[{-6, -7, -6, -1}, {1, -7, 36, -173}, 40] (* G. C. Greubel, Jan 02 2023 *)

Prog

(PARI) Vec((1-x+x^2)/((1+x+x^2)*(1+5*x+x^2)) + O(x^25)) \\ Colin Barker, Apr 30 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+x^2)/((1+x+x^2)*(1+5*x+x^2)) )); // G. C. Greubel, Jan 02 2023
(SageMath)
def U(n, x): return chebyshev_U(n, x)
def A110310(n): return (1/2)*(3*U(n, -5/2) - U(n, -1/2))
[A110310(n) for n in range(41)] # G. C. Greubel, Jan 02 2023

Crossrefs

Cf. A004253, A049347, A109265, A110307, A110308, A110309, A110311.

Sequence in context: A102053 A058681 A246417 * A054493 A368575 A037538

Adjacent sequences: A110307 A110308 A110309 * A110311 A110312 A110313

Keyword

easy,sign

Author

Creighton Dement, Jul 19 2005

Status

approved