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%I A056827 #41 Aug 13 2022 06:23:21
%S A056827 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 A056827 104,112,121,130,140,150,160,170,181,192,204,216,228,240,253,266,280,
%U A056827 294,308,322,337,352,368,384,400,416,433,450,468,486,504
%N A056827 a(n) = floor(n^2/6).
%C A056827 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 A056827 Vincenzo Librandi, <a href="/A056827/b056827.txt">Table of n, a(n) for n = 0..5000</a>
%H A056827 <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 A056827 From _R. J. Mathar_, Nov 22 2008: (Start)
%F A056827 G.f.: x^3*(1+x^2)/((1+x)*(1-x)^3*(1+x+x^2)*(1-x+x^2)).
%F A056827 a(n+1) - a(n) = A123919(n). (End)
%F A056827 a(n) = floor( (1/2) * Sum_{i=1..n+1} (ceiling(i/3) + floor(i/3) - 1) ). - _Wesley Ivan Hurt_, Jun 06 2014
%F A056827 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
%e A056827 5|                                             .__.__.
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%e A056827 0|.__.__./_|__|__|__|__|__|__|__|__|__|__|__|__|__|__|_________________
%e A056827   0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17   .. n
%e A056827   0  0  0  1  2  4  6  8 10 13 16 20 24 28 32 37 42 48   .. a(n)
%e A056827      0  0  0  1  2  4  6  8 10 13 16 20 24 28 32 37 42   .. a(n-1) <--
%p A056827 A056827:=n->floor(n^2/6); seq(A056827(k), k=0..60); # _Wesley Ivan Hurt_, Oct 29 2013
%t A056827 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 A056827 (Magma)[Floor(n^2/6): n in [0..60]]; // _Vincenzo Librandi_, May 08 2011
%o A056827 (PARI) n^2\6 \\ _Charles R Greathouse IV_, May 08 2011
%o A056827 (Sage) [floor(n^2/6) for n in (0..60)] # _G. C. Greubel_, Jul 23 2019
%o A056827 (GAP) List([0..60], n-> Int(n^2/6) ); # _G. C. Greubel_, Jul 23 2019
%Y A056827 Cf. A000290, A007590, A000212, A002620, A118015, A118013, A056834, A130519, A056838, A056865.
%K A056827 nonn,easy
%O A056827 0,5
%A A056827 _N. J. A. Sloane_, Sep 02 2000

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