A056827 a(n) = floor(n^2/6).

0, 0, 0, 1, 2, 4, 6, 8, 10, 13, 16, 20, 24, 28, 32, 37, 42, 48, 54, 60, 66, 73, 80, 88, 96, 104, 112, 121, 130, 140, 150, 160, 170, 181, 192, 204, 216, 228, 240, 253, 266, 280, 294, 308, 322, 337, 352, 368, 384, 400, 416, 433, 450, 468, 486, 504

Offset: 0

Author

N. J. A. Sloane, Sep 02 2000

Keywords

• nonn
• easy

Comments

a(n-1) represents the floor of the area under the polygon connecting the lattice points (n, floor(n/3)) from 0..n, n>0 (see example). - Wesley Ivan Hurt, Jun 06 2014

Examples

			5|                                             .__.__.
.|                                            /|  |  |
4|                                    .__.__./_|__|__|
.|                                   /|  |  |  |  |  |
3|                           .__.__./_|__|__|__|__|__|
.|                          /|  |  |  |  |  |  |  |  |
2|                  .__.__./_|__|__|__|__|__|__|__|__|
.|                 /|  |  |  |  |  |  |  |  |  |  |  |
1|         .__.__./_|__|__|__|__|__|__|__|__|__|__|__|
.|        /|  |  |  |  |  |  |  |  |  |  |  |  |  |  |
0|.__.__./_|__|__|__|__|__|__|__|__|__|__|__|__|__|__|_________________
  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17   .. n
  0  0  0  1  2  4  6  8 10 13 16 20 24 28 32 37 42 48   .. a(n)
     0  0  0  1  2  4  6  8 10 13 16 20 24 28 32 37 42   .. a(n-1) <--
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,1,-2,1).

Crossrefs

Cf. A000290, A007590, A000212, A002620, A118015, A118013, A056834, A130519, A056838, A056865.

Programs

GAP

List([0..60], n-> Int(n^2/6) ); # G. C. Greubel, Jul 23 2019

Magma

[Floor(n^2/6): n in [0..60]]; // Vincenzo Librandi, May 08 2011

Maple

A056827:=n->floor(n^2/6); seq(A056827(k), k=0..60); # Wesley Ivan Hurt, Oct 29 2013

Mathematica

Floor[Range[0,60]^2/6] (* or *) LinearRecurrence[{2,-1,0,0,0,1,-2,1}, {0,0,0,1,2,4,6,8}, 60] (* Harvey P. Dale, Jun 06 2013 *)

PARI

n^2\6 \\ Charles R Greathouse IV, May 08 2011

Sage

[floor(n^2/6) for n in (0..60)] # G. C. Greubel, Jul 23 2019

Formula

From R. J. Mathar, Nov 22 2008: (Start)

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

a(n+1) - a(n) = A123919(n). (End)

a(n) = floor( (1/2) * Sum_{i=1..n+1} (ceiling(i/3) + floor(i/3) - 1) ). - Wesley Ivan Hurt, Jun 06 2014

Sum_{n>=3} 1/a(n) = 15/8 + Pi^2/36 - Pi/(4*sqrt(3)) + tan(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)). - Amiram Eldar, Aug 13 2022