A097123 Expansion of (1-x)^2/((1-x)^3 - 4*x^3).

1, 1, 1, 5, 17, 41, 97, 253, 673, 1745, 4481, 11573, 30001, 77689, 200929, 519725, 1344833, 3479969, 9004033, 23296357, 60276817, 155961545, 403535969, 1044107357, 2701521889, 6989923441, 18085741441, 46795063445, 121077583217

Offset: 0

Author

Paul Barry, Jul 25 2004

Keywords

• easy
• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,5)

Crossrefs

Cf. A097122.

Programs

Mathematica

LinearRecurrence[{3, -3, 5}, {1, 1, 1}, 30] (* Amiram Eldar, Oct 11 2021 *)
CoefficientList[Series[(1-x)^2/((1-x)^3-4x^3),{x,0,30}],x] (* Harvey P. Dale, May 06 2022 *)

PARI

a(n) = sum(k=0, n\3, binomial(n, 3*k) * 4^k); \\ Michel Marcus, Oct 11 2021

Formula

G.f.: (1-2*x+x^2)/(1-3*x+3*x^2-5*x^3).

a(n) = 3*a(n-1) - 3*a(n-2) + 5*a(n-3).

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