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

2,3

Comments

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

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)

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

Example

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

Prog

(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

Crossrefs

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

Sequence in context: A102359 A000397 A200765 * A130410 A202807 A203324

Adjacent sequences: A239570 A239571 A239572 * A239574 A239575 A239576

Keyword

nonn,easy

Author

Heinrich Ludwig, Mar 23 2014

Status

approved