A099783 a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k)*3^(n-2*k).

1, 3, 9, 30, 108, 405, 1548, 5967, 23085, 89451, 346842, 1345248, 5218263, 20242872, 78528609, 304640595, 1181814705, 4584708702, 17785841652, 68998115709, 267670245492, 1038395956527, 4028337876861, 15627474388899, 60624993311226

Offset: 0

Author

Paul Barry, Oct 26 2004

Keywords

• easy
• nonn

Comments

In general a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * u^k * v^(n-3*k) has g.f. (1-v*x)/((1-v*x)^2 - u*x^2) and satisfies the recurrence a(n) = 2*u*v*a(n-1) - v^2*a(n-2) + u*a(n-3).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-9,3).

Crossrefs

Cf. A099780, A099781, A099782, A099784, A099785, A099786, A099787.

Programs

GAP

a:=[1,3,9];; for n in [4..30] do a[n]:=6*a[n-1]-9*a[n-2]+3*a[n-3]; od; a; # G. C. Greubel, Sep 04 2019

Magma

I:=[1,3,9]; [n le 3 select I[n] else 6*Self(n-1) - 9*Self(n-2) + 3*Self(n-3): n in [1..30]]; // G. C. Greubel, Sep 04 2019

Maple

seq(coeff(series((1-3*x)/((1-3*x)^2 - 3*x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Sep 04 2019

Mathematica

LinearRecurrence[{6,-9,3}, {1,3,9}, 30] (* G. C. Greubel, Sep 04 2019 *)

PARI

my(x='x+O('x^30)); Vec((1-3*x)/((1-3*x)^2 - 3*x^3)) \\ G. C. Greubel, Sep 04 2019

Sage

def A099783_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-3*x)/((1-3*x)^2 - 3*x^3)).list()
A099783_list(30) # G. C. Greubel, Sep 04 2019

Formula

G.f.: (1-3*x)/((1-3*x)^2 - 3*x^3).

a(n) = 6*a(n-1) - 9*a(n-2) + 3*a(n-3).