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A056827 [n^2/6]. 15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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

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).

FORMULA

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). - R. J. Mathar, Nov 22 2008

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

EXAMPLE

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) <--

MAPLE

A056827:=n->floor(n^2/6); seq(A056827(k), k=0..100); # 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 *)

PROG

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

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

CROSSREFS

Cf. A000290, A007590, A000212, A002620, A118015, A118013.

Sequence in context: A186347 A302648 A269746 * A024172 A233735 A085577

Adjacent sequences:  A056824 A056825 A056826 * A056828 A056829 A056830

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Sep 02 2000

STATUS

approved

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Last modified December 16 20:51 EST 2018. Contains 318189 sequences. (Running on oeis4.)