A008733 Molien series for 3-dimensional group [2+, n] = 2*(n/2).

1, 0, 2, 1, 4, 2, 6, 4, 9, 6, 12, 9, 16, 12, 20, 16, 25, 20, 30, 25, 36, 30, 42, 36, 49, 42, 56, 49, 64, 56, 72, 64, 81, 72, 90, 81, 100, 90, 110, 100, 121, 110, 132, 121, 144, 132, 156, 144, 169, 156, 182, 169, 196, 182, 210, 196, 225, 210, 240, 225, 256

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,-1,1).

Programs

GAP

List([0..70], n-> Int((n^2 + 5*n + 13 + 3*(n+1)*(-1)^n)/16)); # G. C. Greubel, Jul 30 2019

Magma

[Floor((n^2+5*n+13+3*(n+1)*(-1)^n)/16): n in [0..70]]; // Vincenzo Librandi, Aug 24 2013

Mathematica

CoefficientList[Series[(1+x^3)/((1-x^2)^2*(1-x^4)), {x,0,70}], x] (* Vincenzo Librandi, Aug 24 2013 *)
LinearRecurrence[{1,1,-1,1,-1,-1,1},{1,0,2,1,4,2,6},70] (* Harvey P. Dale, Nov 23 2015 *)

PARI

a(n)=((n^2+5*n+13+3*(n+1)*(-1)^n))\16 \\ Charles R Greathouse IV, Jun 11 2015

Sage

[floor((n^2 + 5*n + 13 + 3*(n+1)*(-1)^n)/16) for n in (0..70)] # G. C. Greubel, Jul 30 2019

Formula

From R. J. Mathar, Nov 04 2008: (Start)

a(n) = A005232(n) - A005232(n-1).

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

a(n) = floor((n^2 + 5*n + 13 + 3*(n+1)*(-1)^n)/16). - Tani Akinari, Aug 23 2013

a(n) = Sum_{i=1..floor((n+4)/2)} floor((i-(n mod 2))/2). - Wesley Ivan Hurt, Mar 31 2014

a(n) = (2*n^2+10*n+13+3*(2*n+5)*(-1)^n+4*(-1)^((6*n-1+(-1)^n)/4))/32. - Luce ETIENNE, Jun 09 2015