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

Author

Paul Barry, Oct 20 2004

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A000071, A000302, A076264, A099504.

Programs

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

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 *)

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

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).