A008806 Expansion of (1+x^3)/((1-x^2)^2*(1-x^3)).

1, 0, 2, 2, 3, 4, 6, 6, 9, 10, 12, 14, 17, 18, 22, 24, 27, 30, 34, 36, 41, 44, 48, 52, 57, 60, 66, 70, 75, 80, 86, 90, 97, 102, 108, 114, 121, 126, 134, 140, 147, 154, 162, 168, 177, 184, 192, 200, 209, 216, 226, 234, 243, 252, 262, 270, 281, 290, 300, 310, 321

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• David Beckwith, Vadim Ponomarenko and Rob Pratt, Building Two Piles of Equal Height: 11183, The American Mathematical Monthly, 114 (2007), 551-552.
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Programs

GAP

a:=[1,0,2,2,3,4];; for n in [7..70] do a[n]:=a[n-1]+a[n-2]-a[n-4]-a[n-5]+a[n-6]; od; a; # G. C. Greubel, Sep 12 2019

Magma

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x^3)/((1-x^2)^2*(1-x^3)) )); // G. C. Greubel, Sep 12 2019

Maple

seq(coeff(series((1+x^3)/((1-x^2)^2*(1-x^3)), x, n+1), x, n), n = 0..70); # G. C. Greubel, Sep 12 2019

Mathematica

CoefficientList[Series[(1+x^3)/((1-x^2)^2*(1-x^3)), {x,0,70}], x] (* or *) LinearRecurrence[{1,1,0,-1,-1,1}, {1,0,2,2,3,4}, 70] (* G. C. Greubel, Sep 12 2019 *)

PARI

Vec((1+x^3)/((1-x^2)^2*(1-x^3)) +O(x^70)) \\ Charles R Greathouse IV, Sep 26 2012; modified by G. C. Greubel, Sep 12 2019

Sage

def A008806_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^3)/((1-x^2)^2*(1-x^3))).list()
A008806_list(70) # G. C. Greubel, Sep 12 2019

Formula

From R. J. Mathar, Nov 08 2010: (Start)

a(n) = (16*A131713(n) +29 +24*n +6*n^2 +27*(-1)^n)/72.

G.f.: (1 -x +x^2)/( (1+x)*(1+x+x^2)*(1-x)^3 ). (End)

a(n) = floor((6*n^2+24*n+61+27*(-1)^n)/72). - Tani Akinari, Jul 24 2013

Extensions

Terms a(52) onward added by G. C. Greubel, Sep 12 2019