A154996 a(n) = 5*a(n-1) + 20*a(n-2), n>2 ; a(0)=1, a(1)=1, a(2)=9.

1, 1, 9, 65, 505, 3825, 29225, 222625, 1697625, 12940625, 98655625, 752090625, 5733565625, 43709640625, 333219515625, 2540290390625, 19365842265625, 147635019140625, 1125491941015625, 8580160087890625, 65410639259765625

Offset: 0

Author

Philippe Deléham, Jan 18 2009

Keywords

• nonn

Comments

The sequences A155001, A155000, A154999, A154997 and A154996 have a common form: a(0)=a(1)=1, a(2)= 2*b+1, a(n) = (b+1)*(a(n-1) + b*a(n-2)), with b some constant. The generating function of these is (1 - b*x - b^2*x^2)/(1 - (b+1)*x - b*(1+b)*x^2). - R. J. Mathar, Jan 20 2009

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,20).

Crossrefs

Cf. A154997, A154999, A155000, A155001.

Programs

Magma

I:=[1,9]; [1] cat [n le 2 select I[n] else 5*(Self(n-1) +4*Self(n-2)): n in [1..30]]; // G. C. Greubel, Apr 21 2021

Maple

m:=30; S:=series( (1-4*x-16*x^2)/(1-5*x-20*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Apr 21 2021

Mathematica

Join[{1},LinearRecurrence[{5,20},{1,9},20]] (* Harvey P. Dale, Jan 19 2012 *)

Sage

def A154996_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-4*x-16*x^2)/(1-5*x-20*x^2) ).list()
A154996_list(30) # G. C. Greubel, Apr 21 2021

Formula

G.f.: (1 -4*x -16*x^2)/(1 -5*x -20*x^2).

a(n+1) = Sum_{k=0..n} A154929(n,k)*4^(n-k).