A110309 Expansion of (1+3*x+x^2)/((1+x+x^2)*(1+5*x+x^2)).

1, -3, 12, -57, 275, -1320, 6325, -30303, 145188, -695637, 3332999, -15969360, 76513801, -366599643, 1756484412, -8415822417, 40322627675, -193197315960, 925663952125, -4435122444663, 21249948271188, -101814618911277, 487823146285199, -2337301112514720

Offset: 0

Author

Creighton Dement, Jul 19 2005

Keywords

• easy
• sign

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).

Crossrefs

Cf. A004253, A049347, A109265, A110307, A110308, A110310, A110311.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+3*x+x^2)/((1+x+x^2)*(1+5*x+x^2)) )); // G. C. Greubel, Jan 03 2023

Maple

seriestolist(series((1+3*x+x^2)/((x^2+5*x+1)*(x^2+x+1)), x=0,25));

Mathematica

LinearRecurrence[{-6,-7,-6,-1}, {1,-3,12,-57}, 40] (* G. C. Greubel, Jan 03 2023 *)

PARI

Vec((1+3*x+x^2)/((1+x+x^2)*(1+5*x+x^2)) + O(x^25)) \\ Colin Barker, Apr 30 2019

SageMath

def A110309(n): return (1/2)*(chebyshev_U(n,-5/2)+chebyshev_U(n,-1/2))
[A110309(n) for n in range(41)] # G. C. Greubel, Jan 03 2023

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)*(ChebyshevU(n, -5/2) + ChebyshevU(n, -1/2)). - G. C. Greubel, Jan 03 2023