A051959 Expansion of (1+6*x)/((1-2*x-x^2)*(1-x)^2).

1, 10, 36, 104, 273, 686, 1688, 4112, 9969, 24114, 58268, 140728, 339809, 820438, 1980784, 4782112, 11545121, 27872474, 67290196, 162453000, 392196337, 946845822, 2285888136, 5518622256, 13323132817, 32164888066, 77652909132, 187470706520, 452594322369, 1092659351462, 2637913025504, 6368485402688

Offset: 0

Author

Barry E. Williams, Jan 04 2000

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• A. F. Horadam, Special Properties of the Sequence W(n){a,b; p,q}, Fib. Quart., Vol. 5, No. 5 (1967), pp. 424-434.
• Index entries for linear recurrences with constant coefficients, signature (4,-4,0,1).

Crossrefs

Cf. A000129, A048771, A048695.

Programs

Magma

I:=[1,10,36,104]; [n le 4 select I[n] else 4*Self(n-1)-4*Self(n-2) +Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 22 2012

Mathematica

LinearRecurrence[{4,-4,0,1},{1,10,36,104},40] (* Vincenzo Librandi, Jun 22 2012 *)

SageMath

def A051959(n):
    @CachedFunction
    def a(n):
        if n<4: return (1,10,36,104)[n]
        else: return 4*a(n-1) -4*a(n-2) +a(n-4)
    return a(n)
[A051959(n) for n in range(41)] # G. C. Greubel, Nov 11 2024

Formula

a(n) = 2*a(n-1) + a(n-2) + (7*n+1), with a(0)=1, a(1)=10.

a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4).

a(n) = ( (25 + 17*sqrt(2))*(1+sqrt(2))^n - (25 - 17*sqrt(2))*(1-sqrt(2))^n )/(4*sqrt(2)) - (7*n + 15)/2.

a(n) = (1/2)*(4*Pell(n+2) - 3*Pell(n) - 7*n - 15), with Pell(n) = A000129(n). - Ralf Stephan, May 15 2007

E.g.f.: (1/4)*exp(x)*(-30 - 14*x + 25*sqrt(2)*sinh(sqrt(2)*x) + 34*cosh(sqrt(2)*x)). - G. C. Greubel, Nov 11 2024