A169592 Expansion of 1/((1-x)*(1-x^2-x^4)) + x/(1-3*x^3).

1, 2, 2, 2, 7, 4, 7, 16, 12, 12, 47, 20, 33, 114, 54, 54, 331, 88, 143, 872, 232, 232, 2563, 376, 609, 7170, 986, 986, 21279, 1596, 2583, 61632, 4180, 4180, 183911, 6764, 10945, 542386, 17710, 17710, 1622979, 28656, 46367, 4829336, 75024, 75024, 14470299, 121392

Offset: 0

Author

Roger L. Bagula, Dec 02 2009

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 303.
• Index entries for linear recurrences with constant coefficients, signature (1,1,2,-2,-4,3,-3,3).

Crossrefs

Cf. A103609.

Programs

Magma

R:=PowerSeriesRing(Integers(), 60);
Coefficients(R!( 1/((1-x)*(1-x^2-x^4)) +x/(1-3*x^3) )); // G. C. Greubel, Oct 23 2024

Mathematica

p[t_]= 1/((1-t)*(1-t^2-t^4)) + t/(1-3*t^3);
CoefficientList[ Series[p[t], {t, 0, 60}], t]

SageMath

def A169592_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)*(1-x^2-x^4)) +x/(1-3*x^3) ).list()
A169592_list(60) # G. C. Greubel, Oct 23 2024

Formula

G.f.: (1+x+x^4-x^5+x^6-x^2-4*x^3)/((1-x)*(1-3*x^3)*(1-x^2-x^4)).

a(n) = +a(n-1) +a(n-2) +2*a(n-3) -2*a(n-4) -4*a(n-5) +3*a(n-6) -3*a(n-7) 3*a(n-8).

a(n) = A103609(n+6) - 1 + 3^((n-1)/3) if n == 1 mod 3.

a(n) = A103609(n+6) - 1 if n == 0 or 2 mod 3.

Extensions

Notation adapted to OEIS standards, sequence extended, formulas added by the Assoc. Editors of the OEIS [Dec 05 2009]