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

0,3

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).

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

Mathematica

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

Prog

(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
(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

Crossrefs

Cf. A001519, A080937.

Sequence in context: A307787 A221586 A258312 * A005317 A126566 A112808

Adjacent sequences: A123017 A123018 A123019 * A123021 A123022 A123023

Keyword

nonn,easy

Author

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

Extensions

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

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

Status

approved