A239568 Number of ways to place 2 points on a triangular grid of side n so that they are not adjacent.

0, 6, 27, 75, 165, 315, 546, 882, 1350, 1980, 2805, 3861, 5187, 6825, 8820, 11220, 14076, 17442, 21375, 25935, 31185, 37191, 44022, 51750, 60450, 70200, 81081, 93177, 106575, 121365, 137640, 155496, 175032, 196350, 219555, 244755, 272061, 301587, 333450, 367770

Offset: 2

Author

Heinrich Ludwig, Mar 22 2014

Keywords

• nonn
• easy

Links

• Heinrich Ludwig, Table of n, a(n) for n = 2..1000
• M. J. Hay, J. Schiff, N. J. Fisch, Maximal energy extraction under discrete diffusive exchange, arXiv preprint arXiv:1508.03499, 2015
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Crossrefs

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

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

Programs

PARI

concat(0, Vec(3*x^3*(x-2)/(x-1)^5 + O(x^100))) \\ Colin Barker, Mar 22 2014

Formula

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

G.f.: 3*x^3*(x-2) / (x-1)^5. - Colin Barker, Mar 22 2014

a(n) = Sum_{i=0..n} (i+5)*A000217(i). - Bruno Berselli, Apr 29 2014

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