A099524 Expansion of 1/(1-5*x-x^3).

1, 5, 25, 126, 635, 3200, 16126, 81265, 409525, 2063751, 10400020, 52409625, 264111876, 1330959400, 6707206625, 33800145001, 170331684405, 858365628650, 4325628288251, 21798473125660, 109850731256950, 553579284573001

Offset: 0

Author

Paul Barry, Oct 20 2004

Keywords

• nonn
• easy

Comments

A transform of A000351 under the mapping mapping g(x)->(1/(1-x^3))g(x/(1-x^3)).

a(n) equals the number of n-length words on {0,1,2,3,4,5} such that 0 appears only in a run which length is a multiple of 3. - Milan Janjic, Feb 17 2015

Links

• Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3
• Index entries for linear recurrences with constant coefficients, signature (5,0,1).

Crossrefs

Cf. A000351, A099504.

Programs

Mathematica

CoefficientList[Series[1/(1-5x-x^3),{x,0,30}],x] (* or *) LinearRecurrence[ {5,0,1},{1,5,25},30] (* Harvey P. Dale, May 08 2012 *)

Formula

a(n) = 5*a(n-1) + a(n-3).

a(n) = Sum_(k=0..floor(n/3)) binomial(n-2*k, k)*5^(n-3*k).