A092353 Expansion of (1+x^3)/((1-x)^2*(1-x^3)^2).

1, 2, 3, 7, 11, 15, 24, 33, 42, 58, 74, 90, 115, 140, 165, 201, 237, 273, 322, 371, 420, 484, 548, 612, 693, 774, 855, 955, 1055, 1155, 1276, 1397, 1518, 1662, 1806, 1950, 2119, 2288, 2457, 2653, 2849, 3045, 3270, 3495, 3720, 3976, 4232, 4488, 4777, 5066, 5355, 5679

Offset: 0

Author

N. J. A. Sloane, Mar 20 2004

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Crossrefs

Cf. A005993.

Programs

Maple

seq(add(floor(i/3)^2, i=1..n+3), n=0..60); # Ridouane Oudra, Oct 19 2019

Mathematica

a[n_] := Sum[Floor[i/3]^2, {i,1,n+3}]; Table[a[n], {n, 0, 100}] (* Enrique Pérez Herrero, Mar 20 2012 *)

Sage

def A092353():
    a, b, c, m = 0, 0, 0, 0
    while True:
        yield (a*(a*(2*a+9)+13)+b*(b+1)*(2*b+1)+c*(c+1)*(2*c+1)+6)//6
        m = m + 1 if m < 2 else 0
        if   m == 0: a += 1
        elif m == 1: b += 1
        elif m == 2: c += 1
a = A092353()
print([next(a) for  in range(52)]) # _Peter Luschny, May 04 2016

Formula

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

a(n) = Sum(i=1..n+3, floor(i/3)^2). - Enrique Pérez Herrero, Mar 20 2012

a(n) = (1/2)*(-4*t^3 + (2n-7)*t^2 + (4n-1)*t +2n +2), where t = floor(n/3). - Ridouane Oudra, Oct 19 2019