A117899 Expansion of (1 + 2*x + 5*x^2 + 3*x^3 + 2*x^4)/(1-x^3)^2.

1, 2, 5, 5, 6, 10, 9, 10, 15, 13, 14, 20, 17, 18, 25, 21, 22, 30, 25, 26, 35, 29, 30, 40, 33, 34, 45, 37, 38, 50, 41, 42, 55, 45, 46, 60, 49, 50, 65, 53, 54, 70, 57, 58, 75, 61, 62, 80, 65, 66, 85, 69, 70, 90, 73, 74, 95, 77, 78, 100, 81, 82, 105, 85, 86, 110, 89, 90, 115, 93

Offset: 0

Author

Paul Barry, Apr 01 2006

Keywords

• easy
• nonn

Comments

Row sums of A117898.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Crossrefs

Cf. A117898.

Programs

Magma

I:=[1,2,5,5,6,10]; [n le 6 select I[n] else 2*Self(n-3) - Self(n-6): n in [1..91]]; // G. C. Greubel, Oct 01 2021

Mathematica

CoefficientList[Series[(1+2x+5x^2+3x^3+2x^4)/(1-x^3)^2,{x,0,90}],x] (* or *) LinearRecurrence[{0,0,2,0,0,-1},{1,2,5,5,6,10},90] (* Harvey P. Dale, Dec 18 2013 *)

Sage

def A117899_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+2*x+5*x^2+3*x^3+2*x^4)/(1-x^3)^2 ).list()
A117899_list(90) # G. C. Greubel, Oct 01 2021

Formula

a(n) = 2*a(n-3) - a(n-6).

a(n) = Sum_{k=0..n} 2^abs(L(C(n,2)/3) - L(C(k,2)/3)), L(j/p) the Legendre symbol of j and p.