A099782 a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * 2^k * 4^(n-3*k).

1, 4, 16, 66, 280, 1216, 5380, 24144, 109504, 500488, 2300128, 10612224, 49096720, 227578432, 1056304384, 4907373600, 22813275520, 106100835328, 493609021504, 2296885357824, 10689540189184, 49753373831296, 231588118339072

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,-16,2).

Crossrefs

Cf. A099780, A099781, A099783, A099784, A099785, A099786, A099787.

Programs

GAP

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

Maple

seq(coeff(series((1-4*x)/((1-4*x)^2 - 2*x^3), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Sep 04 2019

Mathematica

LinearRecurrence[{8,-16,2}, {1,4,16}, 30] (* G. C. Greubel, Sep 04 2019 *)

PARI

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

Sage

def A099782_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-4*x)/((1-4*x)^2 - 2*x^3)).list()
A099782_list(30) # G. C. Greubel, Sep 04 2019

Formula

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

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