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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.

%I A239575 #33 Oct 10 2017 22:34:52
%S A239575 0,0,7,176,1976,12565,57275,207018,634166,1711262,4181915,9428657,
%T A239575 19892816,39684027,75473209,137721045,242391212,413215132,684733527,
%U A239575 1106194950,1746637600,2701244609,4099429895,6114748948,8977257362,12988406970,18539308619,26132434991
%N 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.
%C A239575 Rotations and reflections of placements are not counted. If they are to be counted see A239571.
%H A239575 Heinrich Ludwig, <a href="/A239575/b239575.txt">Table of n, a(n) for n = 3..1000</a>
%H A239575 <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 A239575 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 A239575 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 A239575 There are a(5) = 7 non-equivalent ways to place 5 points (x) on a triangular grid of side 5. These are:
%e A239575         x             x             .             x
%e A239575        . .           . .           . .           . .
%e A239575       x . x         x . x         x . x         . x .
%e A239575      . . . .       . . . .       . . . .       . . . .
%e A239575     x . . . x     . x . x .     x . x . x     x . x . x
%e A239575 .
%e A239575         x             x             x
%e A239575        . .           . .           . .
%e A239575       . x .         . x .         x . x
%e A239575      x . . x       x . . .       . . . .
%e A239575     . . x . .     . . x . x     x . . x .
%t A239575 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 A239575 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 A239575 Cf. A239572, A239571, A032091 (2 points), A239573 (3 points), A239574 (4 points), 279446 (6 points).
%K A239575 nonn,easy
%O A239575 3,3
%A A239575 _Heinrich Ludwig_, Mar 23 2014