A364705 Expansion of 1/(1 - 4*x - x^2 + x^3).

1, 4, 17, 71, 297, 1242, 5194, 21721, 90836, 379871, 1588599, 6643431, 27782452, 116184640, 485877581, 2031912512, 8497342989, 35535406887, 148607058025, 621466295998, 2598936835130, 10868606578493, 45451896853104, 190077257155779, 794892318897727, 3324194635893583

Offset: 0

Author

G. C. Greubel, Aug 04 2023

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,1,-1).

Crossrefs

Cf. A124807, A126393.

Programs

Magma

I:=[1,4,17]; [n le 3 select I[n] else 4*Self(n-1) +Self(n-2) -Self(n-3): n in [1..41]];

Mathematica

LinearRecurrence[{4,1,-1}, {1,4,17}, 41]

SageMath

@CachedFunction
def a(n): # a = A364705
    if (n<3): return (1,4,17)[n]
    else: return 4*a(n-1) +a(n-2) -a(n-3)
[a(n) for n in range(41)]

Formula

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

a(n) = 4*a(n-1) + a(n-2) - a(n-3).

a(n) = Sum_{k=0..n} Sum_{j=0..k} binomial(n-k, j)*binomial(n-k, k-j)*4^(n-2*k)*((1-sqrt(17))/2)^(k-j)*((1+sqrt(17))/2)^j.