A117900 Expansion of (1 + 2*x + 4*x^2 + 4*x^3 + 2*x^4)/((1+x)*(1-x^3)^2).

1, 1, 3, 3, 3, 5, 6, 4, 8, 8, 6, 10, 11, 7, 13, 13, 9, 15, 16, 10, 18, 18, 12, 20, 21, 13, 23, 23, 15, 25, 26, 16, 28, 28, 18, 30, 31, 19, 33, 33, 21, 35, 36, 22, 38, 38, 24, 40, 41, 25, 43, 43, 27, 45, 46, 28, 48, 48, 30, 50, 51, 31, 53, 53, 33, 55, 56, 34, 58, 58, 36

Offset: 0

Author

Paul Barry, Apr 01 2006

Keywords

• easy
• nonn

Comments

Diagonal 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 (-1,0,2,2,0,-1,-1).

Crossrefs

Cf. A117898.

Programs

Magma

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+2*x+4*x^2+4*x^3+2*x^4)/((1+x)*(1-x^3)^2) )); // G. C. Greubel, Oct 01 2021

Mathematica

CoefficientList[Series[(1+2x+4x^2+4x^3+2x^4)/((1-x^3)(1+x-x^3-x^4)),{x,0,80}],x] (* or *) LinearRecurrence[{-1,0,2,2,0,-1,-1},{1,1,3,3,3,5,6},80] (* Harvey P. Dale, Mar 06 2018 *)

Sage

def A117899_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+2*x+4*x^2+4*x^3+2*x^4)/((1+x)*(1-x^3)^2) ).list()
A117899_list(80) # G. C. Greubel, Oct 01 2021

Formula

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

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