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A097122
Expansion of (1-x)^2/((1-x)^3 - 3*x^3).
4
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
0,4
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.
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.
MATHEMATICA
CoefficientList[Series[(1-x)^2/((1-x)^3-3x^3), {x, 0, 40}], x]
PROG
(PARI) a(n) = sum(k=0, n\3, binomial(n, 3*k) * 3^k); \\ Michel Marcus, Oct 11 2021
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jul 25 2004
STATUS
approved