A110307 Expansion of (1+2*x)/((1+x+x^2)*(1+5*x+x^2)).

1, -4, 17, -80, 384, -1842, 8827, -42292, 202631, -970862, 4651680, -22287540, 106786021, -511642564, 2451426797, -11745491420, 56276030304, -269634660102, 1291897270207, -6189851690932, 29657361184451, -142096954231322, 680827409972160, -3262040095629480

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. A002320, A004253, A049347, A099837, A110308, A110309, A110310, A110311.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+2*x)/((1+x+x^2)*(1+5*x+x^2)) )); // G. C. Greubel, Jan 03 2023

Maple

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

Mathematica

LinearRecurrence[{-6,-7,-6,-1}, {1,-4,17,-80}, 41] (* G. C. Greubel, Jan 03 2023 *)

PARI

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

SageMath

def U(n,x): return chebyshev_U(n, x)
def A110307(n): return (1/4)*(3*U(n,-5/2) +U(n-1,-5/2) +U(n,-1/2) -U(n-1,-1/2))
[A110307(n) for n in range(41)] # G. C. Greubel, Jan 03 2023

Formula

a(n+2) = - 5*a(n+1) - a(n) - A099837(n+1).

a(n) + a(n+1) + a(n+2) = A002320(n).

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) = (1/4)*(3*U(n,-5/2) + U(n-1,-5/2) + U(n,-1/2) - U(n-1,-1/2)), where U(n, x) = ChebyshevU(n, x). - G. C. Greubel, Jan 03 2023