%I #33 Oct 10 2017 22:34:52

%S 0,0,7,176,1976,12565,57275,207018,634166,1711262,4181915,9428657,

%T 19892816,39684027,75473209,137721045,242391212,413215132,684733527,

%U 1106194950,1746637600,2701244609,4099429895,6114748948,8977257362,12988406970,18539308619,26132434991

%N 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.

%C Rotations and reflections of placements are not counted. If they are to be counted see A239571.

%H Heinrich Ludwig, <a href="/A239575/b239575.txt">Table of n, a(n) for n = 3..1000</a>

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (6,-10,-10,50,-34,-66,110,0,-110,66,34,-50,10,10,-6,1)

%F 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.

%F 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

%e There are a(5) = 7 non-equivalent ways to place 5 points (x) on a triangular grid of side 5. These are:

%e x x . x

%e . . . . . . . .

%e x . x x . x x . x . x .

%e . . . . . . . . . . . . . . . .

%e x . . . x . x . x . x . x . x x . x . x

%e .

%e x x x

%e . . . . . .

%e . x . . x . x . x

%e x . . x x . . . . . . .

%e . . x . . . . x . x x . . x .

%t 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 *)

%t 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 *)

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

%K nonn,easy

%O 3,3

%A _Heinrich Ludwig_, Mar 23 2014