A097596 Expansion of g.f. x*(1-x+x^5+x^6-x^7+x^9)/(1-2*x+x^4+x^6-2*x^7+x^10).

1, 1, 2, 4, 7, 14, 26, 48, 89, 165, 304, 559, 1029, 1893, 3482, 6404, 11779, 21666, 39850, 73296, 134813, 247961, 456072, 838847, 1542881, 2837801, 5219530, 9600212, 17657543, 32477286, 59735042, 109869872, 202082201, 371687117

Offset

1,3

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,0,0,-1,0,-1,2,0,0,-1).

Formula

a(n) = ((r3)^n - (r2)^n - (r1)^n)/(r3 - r2 - r1), where r3 = (1/3)*(1 + (19 - 3*sqrt(33))^(1/3) + (19 + 3*sqrt(33))^(1/3)), r1 and r2 are the other roots of x^3 - x^2 - x - 1 = 0.

G.f.: x*(1 -x +x^5 +x^6 -x^7 +x^9)/((1-x)*(1+x^2)*(1-x-x^2-x^3)*(1-x^2+x^4)). -Colin Barker, Dec 02 2012

a(n) = 2*a(n-1) - a(n-4) - a(n-6) + 2*a(n-7) - a(n-10). - G. C. Greubel, Dec 06 2022

Mathematica

r1=-0.419643377607080569`-0.606290729207199419` I;
r2=-0.419643377607080569`+0.606290729207199419` I;
r3=1.83928675521416113`;
f[n_]:=(r3^n-r2^n-r1^n)/(r3-r2-r1);
Table[Floor[f[n]], {n, 50}]
LinearRecurrence[{2, 0, 0, -1, 0, -1, 2, 0, 0, -1}, {1, 1, 2, 4, 7, 14, 26, 48, 89, 165}, 50] (* G. C. Greubel, Dec 06 2022 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 51); Coefficients(R!( x*(1-x+x^5+x^6-x^7+x^9)/(1-2*x+x^4+x^6-2*x^7+x^10) )); // G. C. Greubel, Dec 06 2022
(SageMath)
def A097596_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( x*(1-x+x^5+x^6-x^7+x^9)/(1-2*x+x^4+x^6-2*x^7+x^10) ).list()
a=A097596_list(50); a[1:] # G. C. Greubel, Dec 06 2022

Crossrefs

Cf. A001644.

Sequence in context: A088813 A347780 A351013 * A054191 A347761 A257792

Adjacent sequences: A097593 A097594 A097595 * A097597 A097598 A097599

Keyword

nonn,easy

Author

Roger L. Bagula, Sep 20 2004

Extensions

Edited by G. C. Greubel, Dec 06 2022

Status

approved