A123020 Expansion of (1 -5*x +5*x^2)/((1 -2*x)*(1 -4*x +x^2)).

1, 1, 2, 5, 14, 43, 142, 493, 1766, 6443, 23750, 88045, 327406, 1219531, 4546622, 16958765, 63272054, 236096683, 881049142, 3287968813, 12270563966, 45793762763, 170903438510, 637817894125, 2380363943686, 8883629492011

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Sep 24 2006

Keywords

• nonn
• easy

Comments

Denominator of reduced g.f. is essentially the characteristic polynomial of [1, 1, 0; 1, 2, 1; 0, 1, 3]. - Paul Barry, Dec 17 2009

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials
• Index entries for linear recurrences with constant coefficients, signature (6,-9,2).

Crossrefs

Cf. A001519, A080937.

Programs

Magma

I:=[1,1,2]; [n le 3 select I[n] else 6*Self(n-1) - 9*Self(n-2) +2*Self(n-3): n in [1..31]]; // G. C. Greubel, Jul 11 2021

Mathematica

Table[(2^n - ChebyshevT[n + 1, 2] + 4*ChebyshevT[n, 2])/3, {n,0,30}] (* G. C. Greubel, Jul 11 2021 *)

Sage

def a(n): return (1/3)*(2^n - chebyshev_T(n+1, 2) + 4*chebyshev_T(n, 2))
[a(n) for n in (0..30)] # G. C. Greubel, Jul 11 2021

Formula

From Paul Barry, Dec 17 2009: (Start)

G.f.: 1/(1 -x -x^2/(1 -2*x -x^2/(1-3*x))) = (1-5*x+5*x^2)/(1-6*x+9*x^2-2*x^3).

a(n) = ((2+sqrt(3))/6)*(2-sqrt(3))^n + ((2-sqrt(3))/6)*(2+sqrt(3))^n + 2^n/3. (End)

a(n) = (1/3)*(2^n - ChebyshevT(n+1, 2) + 4*ChebyshevT(n, 2)). - G. C. Greubel, Jul 11 2021

3*a(n) = 2^n +A001075(n-1), n>=1. - R. J. Mathar, Aug 05 2021

Extensions

Edited by N. J. A. Sloane, Jun 13 2007

New name and change of offset by G. C. Greubel, Jul 11 2021