A239575 Number of non-equivalent (mod D_3) ways to place 5 indistinguishable points on a triangular grid of side n so that no two of them are adjacent.

0, 0, 7, 176, 1976, 12565, 57275, 207018, 634166, 1711262, 4181915, 9428657, 19892816, 39684027, 75473209, 137721045, 242391212, 413215132, 684733527, 1106194950, 1746637600, 2701244609, 4099429895, 6114748948, 8977257362, 12988406970, 18539308619, 26132434991

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 A239571.

Examples

			There are a(5) = 7 non-equivalent ways to place 5 points (x) on a triangular grid of side 5. These are:
        x             x             .             x
       . .           . .           . .           . .
      x . x         x . x         x . x         . x .
     . . . .       . . . .       . . . .       . . . .
    x . . . x     . x . x .     x . x . x     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 = 3..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-10,-10,50,-34,-66,110,0,-110,66,34,-50,10,10,-6,1)

Crossrefs

Cf. A239572, A239571, A032091 (2 points), A239573 (3 points), A239574 (4 points), 279446 (6 points).

Programs

Mathematica

Table[(n^10 + 5*n^9 - 130*n^8 - 310*n^7 + 7465*n^6 - 1336*n^5 - 202980*n^4 + 464160*n^3 + 1783424*n^2 - 8360064*n + 9192960)/23040 + (1-(-1)^n)/2*(25*n^4 - 94*n^3 - 418*n^2 + 2053*n - 1779)/1536,{n,3,20}] (* Vaclav Kotesovec after Heinrich Ludwig, Mar 31 2014 *)
Drop[CoefficientList[Series[x^2*(-19 - (19 - 114*x + 190*x^2 + 197*x^3 - 816*x^4 + 1636*x^5 + 3793*x^6 + 965*x^7 + 216*x^8 + 194*x^9 - 2278*x^10 + 53*x^11 + 1547*x^12 - 336*x^13 - 351*x^14 + 125*x^15) / ((-1+x)^11*(1+x)^5)), {x, 0, 20}], x], 3] (* Vaclav Kotesovec, Mar 31 2014 *)

Formula

a(n) = (n^10 + 5*n^9 - 130*n^8 - 310*n^7 + 7465*n^6 - 1336*n^5 - 202980*n^4 + 464160*n^3 + 1783424*n^2 - 8360064*n + 9192960)/23040 + IF(MOD(n,2) = 1)*(25*n^4 - 94*n^3 - 418*n^2 + 2053*n - 1779)/1536.

G.f.: x^2*(-19 - (19 - 114*x + 190*x^2 + 197*x^3 - 816*x^4 + 1636*x^5 + 3793*x^6 + 965*x^7 + 216*x^8 + 194*x^9 - 2278*x^10 + 53*x^11 + 1547*x^12 - 336*x^13 - 351*x^14 + 125*x^15) / ((-1+x)^11 * (1+x)^5)). - Vaclav Kotesovec, Mar 31 2014