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A056827 a(n) = floor(n^2/6). 15

%I

%S 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,

%T 104,112,121,130,140,150,160,170,181,192,204,216,228,240,253,266,280,

%U 294,308,322,337,352,368,384,400,416,433,450,468,486,504

%N a(n) = floor(n^2/6).

%C 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

%H Vincenzo Librandi, <a href="/A056827/b056827.txt">Table of n, a(n) for n = 0..5000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,1,-2,1).

%F From _R. J. Mathar_, Nov 22 2008: (Start)

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

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

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

%e 5| .__.__.

%e .| /| | |

%e 4| .__.__./_|__|__|

%e .| /| | | | | |

%e 3| .__.__./_|__|__|__|__|__|

%e .| /| | | | | | | | |

%e 2| .__.__./_|__|__|__|__|__|__|__|__|

%e .| /| | | | | | | | | | | |

%e 1| .__.__./_|__|__|__|__|__|__|__|__|__|__|__|

%e .| /| | | | | | | | | | | | | | |

%e 0|.__.__./_|__|__|__|__|__|__|__|__|__|__|__|__|__|__|_________________

%e 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 .. n

%e 0 0 0 1 2 4 6 8 10 13 16 20 24 28 32 37 42 48 .. a(n)

%e 0 0 0 1 2 4 6 8 10 13 16 20 24 28 32 37 42 .. a(n-1) <--

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

%t 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 *)

%o (MAGMA)[Floor(n^2/6): n in [0..60]]; // _Vincenzo Librandi_, May 08 2011

%o (PARI) n^2\6 \\ _Charles R Greathouse IV_, May 08, 2011

%o (Sage) [floor(n^2/6) for n in (0..60)] # _G. C. Greubel_, Jul 23 2019

%o (GAP) List([0..60], n-> Int(n^2/6) ); # _G. C. Greubel_, Jul 23 2019

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

%K nonn,easy

%O 0,5

%A _N. J. A. Sloane_, Sep 02 2000

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Last modified October 14 09:25 EDT 2019. Contains 327995 sequences. (Running on oeis4.)