A248088 a(n) = Sum_{k=0..floor(n/4)} binomial(n-3k, k)*(-3)^(3k)*4^(n-4k).

1, 4, 16, 64, 229, 808, 2800, 9472, 31705, 105004, 344416, 1121920, 3631645, 11691472, 37466656, 119574784, 380244721, 1205309140, 3809636848, 12010028224, 37773505429, 118550674936, 371342504848, 1161099257344, 3624512382793, 11297181307900, 35162477600704

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

P. S. Bruckman and G. C. Greubel, Advanced Problem H-725, Fibonacci Quarterly, 52(2):187-190, 2014.

Index entries for linear recurrences with constant coefficients, signature (4,0,0,-27).

Formula

a(n) = 3^n*(3*n + 5)/6 - (-1)^n*3^(n+1)/2*sin((n-1)*arcsin(sqrt(2/3)))/(6*sqrt(2)).

G.f.: 1/(1 - 4x + 27x^4).

a(n) = (1+3/n)*a(n-1) + (3+6/n)*a(n-2) + (9+9/n)*a(n-3). - Robert Israel, Oct 27 2014

Maple

gser := series(1/(1-4*x+27*x^4), x = 0, 35): seq(coeff(gser, x, n), n = 0 .. 30);
#Alternative:
F:= gfun[rectoproc]({(n+4)*a(n+4)+(-7-n)*a(n+3)+(-18-3*n)*a(n+2)+(-45-9*n)*a(n+1), a(0) = 1, a(1) = 4, a(2) = 16, a(3) = 64}, a(n), remember):
seq(F(n), n=0..100); # Robert Israel, Oct 27 2014

Mathematica

CoefficientList[Series[1/(1 - 4 x + 27 x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 28 2014 *)

Prog

(PARI) Vec(1/(1-4*x+27*x^4)+O(x^99)) \\ Charles R Greathouse IV, Oct 28 2014

Crossrefs

Sequence in context: A262334 A065738 A248089 * A294037 A228735 A289694

Adjacent sequences: A248085 A248086 A248087 * A248089 A248090 A248091

Keyword

nonn,easy

Author

Emeric Deutsch, Oct 27 2014

Status

approved