A097122 Expansion of (1-x)^2/((1-x)^3 - 3*x^3).

1, 1, 1, 4, 13, 31, 70, 169, 421, 1036, 2521, 6139, 14998, 36661, 89545, 218644, 533941, 1304071, 3184966, 7778449, 18996733, 46394716, 113307745, 276726019, 675833686, 1650553981, 4031064961, 9844867684, 24043624093, 58720529071

Offset: 0

Author

Paul Barry, Jul 25 2004

Keywords

• easy
• nonn

References

• Maribel Díaz Noguera [Maribel Del Carmen Díaz Noguera], Rigoberto Flores, Jose L. Ramirez, and Martha Romero Rojas, Catalan identities for generalized Fibonacci polynomials, Fib. Q., 62:2 (2024), 100-111. See Table 3.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,4).

Crossrefs

Cf. A024493, A052101, A100136.

Programs

Mathematica

CoefficientList[Series[(1-x)^2/((1-x)^3-3x^3),{x,0,40}],x]

PARI

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

Formula

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

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

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