A099786 a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k)*3^(n-4*k).

1, 3, 9, 27, 82, 255, 819, 2727, 9397, 33312, 120537, 441855, 1631017, 6036879, 22345074, 82589247, 304612975, 1120960983, 4116353265, 15088372416, 55224373105, 201895801851, 737506551321, 2692518758163, 9826402960882

Offset: 0

Author

Paul Barry, Oct 26 2004

Keywords

• easy
• nonn

Comments

In general a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k) * u^k * v^(n-4*k) has g.f. (1-v*x)^2/((1-v*x)^3 - u*x^4) and satisfies the recurrence a(n) = 3*v*a(n-1) - 3*v^2*a(n-2) + v^3*a(n-3) + u*a(n-4).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (9,-27,27,1).

Crossrefs

Cf. A003522, A097119.

Cf. A099780, A099781, A099782, A099783, A099784, A099785, A099787.

Programs

GAP

a:=[1,3,9,27];; for n in [5..30] do a[n]:=9*a[n-1]-27*a[n-2] + 27*a[n-3] +a[n-4]; od; a; # G. C. Greubel, Sep 04 2019

Magma

I:=[1,3,9,27]; [n le 4 select I[n] else 9*Self(n-1) - 27*Self(n-2) + 27*Self(n-3) +Self(n-4): n in [1..30]]; // G. C. Greubel, Sep 04 2019

Maple

seq(coeff(series((1-3*x)^2/((1-3*x)^3 - x^4), x, n+1), x, n), n = 0..30); # G. C. Greubel, Sep 04 2019

Mathematica

LinearRecurrence[{9,-27,27,1},{1,3,9,27},40] (* or *) CoefficientList[ Series[-((1-3 x)^2/(x (x (x (x+27)-27)+9)-1)),{x,0,40}],x] (* Harvey P. Dale, Jun 06 2011 *)

PARI

my(x='x+O('x^30)); Vec((1-3*x)^2/((1-3*x)^3 - x^4)) \\ G. C. Greubel, Sep 04 2019

Sage

def A099786_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-3*x)^2/((1-3*x)^3 - x^4)).list()
A099786_list(30) # G. C. Greubel, Sep 04 2019

Formula

G.f.: (1-3*x)^2/((1-3*x)^3 - x^4).

a(n) = 9*a(n-1) - 27*a(n-2) + 27*a(n-3) + a(n-4).