%I #33 Mar 08 2017 11:55:18

%S 0,6,27,75,165,315,546,882,1350,1980,2805,3861,5187,6825,8820,11220,

%T 14076,17442,21375,25935,31185,37191,44022,51750,60450,70200,81081,

%U 93177,106575,121365,137640,155496,175032,196350,219555,244755,272061,301587,333450,367770

%N Number of ways to place 2 points on a triangular grid of side n so that they are not adjacent.

%H Heinrich Ludwig, <a href="/A239568/b239568.txt">Table of n, a(n) for n = 2..1000</a>

%H M. J. Hay, J. Schiff, N. J. Fisch, <a href="http://arxiv.org/abs/1508.03499">Maximal energy extraction under discrete diffusive exchange</a>, arXiv preprint arXiv:1508.03499, 2015

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1)

%F a(n) = n*(n-1)*(n-2)*(n+5)/8.

%F G.f.: 3*x^3*(x-2) / (x-1)^5. - _Colin Barker_, Mar 22 2014

%F a(n) = Sum_{i=0..n} (i+5)*A000217(i). - _Bruno Berselli_, Apr 29 2014

%F a(n) = t(t(n,k),n) + n, where t(n,k) = n*(n+1)/2 + k*n and t(n,1) = A000096(n). - _Bruno Berselli_, Feb 28 2017

%o (PARI) concat(0, Vec(3*x^3*(x-2)/(x-1)^5 + O(x^100))) \\ _Colin Barker_, Mar 22 2014

%Y Cf. A239567, A032091, A239569 (3 points), A239570 (4 points), A239571 (5 points), A282998 (6 points).

%Y Regarding the third formula, see similar sequences listed in A241765.

%K nonn,easy

%O 2,2

%A _Heinrich Ludwig_, Mar 22 2014