A099503 Expansion of 1/(1-4*x+x^3).

1, 4, 16, 63, 248, 976, 3841, 15116, 59488, 234111, 921328, 3625824, 14269185, 56155412, 220995824, 869714111, 3422701032, 13469808304, 53009519105, 208615375388, 820991693248, 3230957253887, 12715213640160, 50039862867392

Offset

0,2

Comments

A transform of A000302 under the mapping g(x) ->(1/(1+x^3)) * g(x/(1+x^3)).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,0,-1).

Formula

a(n) = 4*a(n-1) - a(n-3).

a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k, k)*(-1)^k*4^(n-3*k).

Mathematica

CoefficientList[Series[1/(1-4x+x^3), {x, 0, 30}], x]  (* Harvey P. Dale, Apr 01 2011 *)
LinearRecurrence[{4, 0, -1}, {1, 4, 16}, 30] (* G. C. Greubel, Aug 03 2023 *)

Prog

(Magma) [n le 3 select 4^(n-1) else 4*Self(n-1) -Self(n-3): n in [1..30]]; // G. C. Greubel, Aug 03 2023
(SageMath)
@CachedFunction
def a(n): # a = A099503
    if (n<3): return 4^n
    else: return 4*a(n-1) - a(n-3)
[a(n) for n in range(31)] # G. C. Greubel, Aug 03 2023

Crossrefs

Cf. A000071, A000302, A076264, A099504.

Sequence in context: A071264 A077822 A257838 * A119376 A282310 A022030

Adjacent sequences: A099500 A099501 A099502 * A099504 A099505 A099506

Keyword

easy,nonn

Author

Paul Barry, Oct 20 2004

Status

approved