A099781 a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * 4^(n-3*k).

1, 4, 16, 65, 268, 1120, 4737, 20244, 87280, 379073, 1656348, 7272896, 32060673, 141775396, 628505296, 2791696705, 12419264300, 55315472416, 246607247233, 1100229683508, 4911436984752, 21934428189121, 97992663440444

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 (8,-16,1).

Programs

GAP

a:=[1,4,16];; for n in [4..30] do a[n]:=8*a[n-1]-16*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Sep 04 2019

Magma

I:=[1,4,16]; [n le 3 select I[n] else 8*Self(n-1) - 16*Self(n-2) + Self(n-3): n in [1..30]];

Maple

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

Mathematica

LinearRecurrence[{8,-16,1},{1,4,16},30] (* Harvey P. Dale, Jul 07 2013 *)

PARI

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

Sage

def A099781_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-4*x)/((1-4*x)^2 -x^3)).list()
A099781_list(30) # G. C. Greubel, Sep 04 2019

Formula

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

a(n) = 8*a(n-1) - 16*a(n-2) + a(n-3).