A008732 Molien series for 3-dimensional group [2,n] = *22n.

1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 18, 21, 24, 27, 30, 34, 38, 42, 46, 50, 55, 60, 65, 70, 75, 81, 87, 93, 99, 105, 112, 119, 126, 133, 140, 148, 156, 164, 172, 180, 189, 198, 207, 216, 225, 235, 245, 255, 265

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• tabf

Examples

			From _Philippe Deléham_, Apr 05 2013: (Start)
Stored in five columns:
    1   2   3   4   5
    7   9  11  13  15
   18  21  24  27  30
   34  38  42  46  50
   55  60  65  70  75
   81  87  93  99 105
  112 119 126 133 140
(End)
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 188
• Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 8a, lambda=5]
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Crossrefs

Cf. A130520.

Programs

GAP

List([0..50], n-> Int((n+3)*(n+4)/10)); # G. C. Greubel, Jul 30 2019

Magma

[Floor((n+3)*(n+4)/10): n in [0..50] ]; // Vincenzo Librandi, Aug 21 2011

Maple

A092202 := proc(n) op(1+(n mod 5),[0,1,0,-1,0]) ; end proc:
A010891 := proc(n) op(1+(n mod 5),[1,-1,0,0,0]) ; end proc:
A008732 := proc(n) (n+2)*(n+5)/10+(A010891(n-1)+2*A092202(n-1))/5 ; end proc:

Mathematica

LinearRecurrence[{2, -1, 0, 0, 1, -2, 1}, {1, 2, 3, 4, 5, 7, 9}, 50] (* Jean-François Alcover, Jan 18 2018 *)

PARI

a(n)=(n+3)*(n+4)\10 \\ Charles R Greathouse IV, Oct 07 2015

Sage

[floor((n+3)*(n+4)/10) for n in (0..50)] # G. C. Greubel, Jul 30 2019

Formula

a(n) = floor( (n+3)*(n+4)/10 ) = (n+2)*(n+5)/10 + b(n)/5 where b(n) = A010891(n-2) + 2*A092202(n-1) = 0, 1, 1, 0, -2, ... with period length 5.

G.f.: 1/((1-x)^2*(1-x^5)).

a(n) = a(n-5) + n + 1. - Paul Barry, Jul 14 2004

From Mitch Harris, Sep 08 2008: (Start)

a(n) = Sum_{j=0..n+5} floor(j/5).

a(n-5) = (1/2)floor(n/5)*(2*n - 3 - 5*floor(n/5)). (End)

a(n) = A130520(n+5). - Philippe Deléham, Apr 05 2013

a(5n) = A000566(n+1), a(5n+1) = A005476(n+1), a(5n+2) = A005475(n+1), a(5n+3) = A147875(n+2), a(5n+4) = A028895(n+1); these formulas correspond to the 5 columns of the array shown in example. - Philippe Deléham, Apr 05 2013