A239573 Number of non-equivalent (mod D_3) ways to place 3 indistinguishable points on a triangular grid of side n so that no two of them are adjacent.

0, 1, 6, 32, 113, 329, 790, 1702, 3320, 6057, 10400, 17074, 26903, 41047, 60796, 87886, 124220, 172275, 234732, 314992, 416703, 544391, 702878, 898040, 1136098, 1424521, 1771178, 2185392, 2676947, 3257305, 3938450, 4734286, 5659306, 6730177, 7964228, 9381234

Offset: 2

Author

Heinrich Ludwig, Mar 23 2014

Keywords

• nonn
• easy

Comments

Rotations and reflections of placements are not counted. If they are to be counted see A239569.

Examples

			There are a(4) = 6 non-equivalent ways to place 3 points on a triangular grid of side 4:
    .           X           X           X           X           X
   . X         . .         . .         . .         . .         . .
  X . .       X . X       X . .       X . .       . X .       . . .
. . X .     . . . .     . . X .     . . . X     . . . X     X . . X
		

Links

• Heinrich Ludwig, Table of n, a(n) for n = 2..1000
• Index entries for linear recurrences with constant coefficients, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1)

Crossrefs

Cf. A239572, A239569, A032091 (2 points), A239574 (4 points), A239575 (5 points), A279446 (6 points).

Programs

PARI

concat(0, Vec(-x^3*(2*x^9 +x^8 -8*x^7 -9*x^6 +3*x^5 +29*x^4 +24*x^3 +14*x^2 +3*x +1)/((x -1)^7*(x +1)^3*(x^2 +x +1)) + O(x^100))) \\ Colin Barker, Mar 23 2014

Formula

a(n) = (n^6 + 3*n^5 - 39*n^4 + 10*n^3 + 456*n^2 - 1008*n + 576)/288 + IF(MOD(n, 2) = 1)*(3*n^2 - 5*n - 5)/32 + IF(MOD(n, 3) = 1)*2/9.

G.f.: -x^3*(2*x^9 +x^8 -8*x^7 -9*x^6 +3*x^5 +29*x^4 +24*x^3 +14*x^2 +3*x +1) / ((x -1)^7*(x +1)^3*(x^2 +x +1)). - Colin Barker, Mar 23 2014