A099025 Expansion of 1 / ((1+x) * (1-5*x+x^2)).

1, 4, 20, 95, 456, 2184, 10465, 50140, 240236, 1151039, 5514960, 26423760, 126603841, 606595444, 2906373380, 13925271455, 66719983896, 319674648024, 1531653256225, 7338591633100, 35161304909276, 168467932913279, 807178359657120, 3867423865372320

Offset: 0

Author

Ralf Stephan, Sep 26 2004

Keywords

• nonn
• easy

Examples

			1 + 4*x + 20*x^2 + 95*x^3 + 456*x^4 + 2184*x^5 + 10465*x^6 + ...
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., Vol. 58, No. 2 (2020), 140-142.
• Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
• Index entries for linear recurrences with constant coefficients, signature (4,4,-1).

Crossrefs

First differences of A089927. First differences are in A003769 and A005386. Pairwise sums are in A004254.

Programs

Magma

I:=[1, 4, 20]; [n le 3 select I[n] else 4*Self(n-1) + 4*Self(n-2) - Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 31 2017

Mathematica

CoefficientList[Series[1/((1+x)*(1-5*x+x^2)), {x,0,50}], x] (* or *) LinearRecurrence[{4,4,-1}, {1,4,20}, 30] (* G. C. Greubel, Dec 31 2017 *)

PARI

Vec(1/(1+x)/(1-5*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

PARI

{a(n) = (3 * (-1)^n + 38 * subst( poltchebi(n), x, 5/2) - 8 * subst( poltchebi(n-1), x, 5/2)) / 21} /* Michael Somos, Jan 25 2013 */

Formula

a(n) = (1/7)*[A030221(n+2) - A003501(n+2) + (-1)^n].

a(n) = 5*a(n-1) -a(n-2) +(-1)^n, a(0)=1, a(1)=4. - Vincenzo Librandi, Mar 22 2011

G.f.: 1 / ((1 + x) * (1 - 5*x + x^2)).

a(-3-n) = -a(n). - Michael Somos, Jan 25 2013

a(n) = (2^(-n)*(3*(-2)^n+(9-2*sqrt(21))*(5-sqrt(21))^n+(5+sqrt(21))^n*(9+2*sqrt(21))))/21. - Colin Barker, Nov 02 2016