A239574 Number of non-equivalent (mod D_3) ways to place 4 indistinguishable points on a triangular grid of side n so that no two of them are adjacent.

0, 1, 24, 200, 1053, 3932, 11988, 31298, 73046, 155880, 310046, 581414, 1038634, 1779531, 2942114, 4714412, 7350595, 11184786, 16654116, 24317554, 34886940, 49252544, 68523846, 94062350, 127534794, 170954603, 226748678, 297809946, 387580007, 500113190, 640178710

Offset: 3

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 A239570.

Examples

			There is a(4) = 1 way to place 4 points on a triangular grid of side n = 4:
      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 (2,3,-5,-8,3,19,4,-24,-15,15,24,-4,-19,-3,8,5,-3,-2,1)

Crossrefs

Cf. A239572, A239570, A032091 (2 points), A239573 (3 points), A239575 (5 points), A279446 (6 points).

Programs

Mathematica

Drop[CoefficientList[Series[x^4*(-1 - 22*x - 149*x^2 - 586*x^3 - 1354*x^4 - 2154*x^5 - 2300*x^6 - 1510*x^7 - 259*x^8 + 470*x^9 + 443*x^10 + 70*x^11 - 130*x^12 - 94*x^13 - 10*x^14 + 18*x^15 + 8*x^16) / ((-1+x)^9 * (1+x)^4 * (1+x+x^2)^3), {x, 0, 20}], x],3] (* Vaclav Kotesovec, Mar 29 2014 *)
Table[(n^8+4*n^7-78*n^6-104*n^5+2556*n^4-3152*n^3-27280*n^2+89664*n-78336)/2304 + If[Mod[n,2]==1,(28*n^3-54*n^2-160*n+129)/768,0] + If[Mod[n,3]==1,(n^2+n-14)/18,0],{n,3,20}] (* Vaclav Kotesovec after Heinrich Ludwig, Mar 29 2014 *)

Formula

a(n) = (n^8 +4*n^7 -78*n^6 -104*n^5 +2556*n^4 -3152*n^3 -27280*n^2 +89664*n -78336)/2304 +IF(n == 1 mod 2)*(28*n^3 -54*n^2 -160*n +129)/768 +IF(n == 1 mod 3)*(n^2 +n -14)/18.

G.f.: x^4*(-1 - 22*x - 149*x^2 - 586*x^3 - 1354*x^4 - 2154*x^5 - 2300*x^6 - 1510*x^7 - 259*x^8 + 470*x^9 + 443*x^10 + 70*x^11 - 130*x^12 - 94*x^13 - 10*x^14 + 18*x^15 + 8*x^16) / ((-1+x)^9 * (1+x)^4 * (1+x+x^2)^3). - Vaclav Kotesovec, Mar 29 2014