A282998 Number of ways to place 6 points on a triangular grid of side n so that no two of them are adjacent.

0, 0, 1, 353, 12231, 153194, 1124820, 5893221, 24425212, 85152341, 259805430, 712840480, 1793423456, 4197531636, 9240962666, 19301854131, 38514786780, 73828909906, 136581190475, 244784427831, 426389859697, 723857976770, 1200460734396, 1948846090829, 3102524331336

Offset: 3

Author

Heinrich Ludwig, Feb 26 2017

Keywords

• nonn
• easy

Comments

Rotations and reflections of placements are counted. If they are to be ignored, see A279446.

Examples

			There is a(5) = 1 way to place 6 points on a triangular grid of side n = 5:
        X
       . .
      X . X
     . . . .
    X . X . X
		

Links

• Heinrich Ludwig, Table of n, a(n) for n = 3..1000
• Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Crossrefs

Cf. A279446, A239567, A239568 (2 points), A239569 (3 points), A239570 (4 points), A239571 (5 points).

Programs

Maple

A282998:=n->(n^12 + 6*n^11 - 195*n^10 - 670*n^9 + 17455*n^8 + 13426*n^7 - 836249*n^6 + 1252990*n^5 + 19599884*n^4 - 68542552*n^3 - 131400416*n^2 + 974223360*n - 1308856320)/46080: 0,seq(A282998(n), n=4..30); # Wesley Ivan Hurt, Apr 10 2017

Mathematica

Drop[CoefficientList[Series[(x^5 * (1 + 340 * x + 7720 * x^2 + 21439 * x^3 - 12927 * x^4 - 27265 * x^5 + 28385 * x^6 - 6252 * x^7 - 116 * x^8 - 2365 * x^9 + 1787 * x^10 - 352 * x^11) / (1 - x)^13 ),{x,0,27}],x],3] (* Indranil Ghosh, Feb 26 2017, from the g.f. by Colin Barker *)

PARI

concat(vector(2), Vec(x^5*(1 + 340*x + 7720*x^2 + 21439*x^3 - 12927*x^4 - 27265*x^5 + 28385*x^6 - 6252*x^7 - 116*x^8 - 2365*x^9 + 1787*x^10 - 352*x^11) / (1 - x)^13 + O(x^30))) \\ Colin Barker, Feb 26 2017

Formula

a(n) = (n^12 + 6*n^11 - 195*n^10 - 670*n^9 + 17455*n^8 + 13426*n^7 - 836249*n^6 + 1252990*n^5 + 19599884*n^4 - 68542552*n^3 - 131400416*n^2 + 974223360*n - 1308856320)/46080 for n>=4.

G.f.: x^5*(1 + 340*x + 7720*x^2 + 21439*x^3 - 12927*x^4 - 27265*x^5 + 28385*x^6 - 6252*x^7 - 116*x^8 - 2365*x^9 + 1787*x^10 - 352*x^11) / (1 - x)^13. - Colin Barker, Feb 26 2017