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%I #15 Mar 08 2017 12:14:21
%S 0,1,6,32,113,329,790,1702,3320,6057,10400,17074,26903,41047,60796,
%T 87886,124220,172275,234732,314992,416703,544391,702878,898040,
%U 1136098,1424521,1771178,2185392,2676947,3257305,3938450,4734286,5659306,6730177,7964228,9381234
%N Number of non-equivalent (mod D_3) ways to place 3 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 A239569.
%H Heinrich Ludwig, <a href="/A239573/b239573.txt">Table of n, a(n) for n = 2..1000</a>
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1)
%F a(n) = (n^6 + 3*n^5 - 39*n^4 + 10*n^3 + 456*n^2 - 1008*n + 576)/288 + IF(MOD(n, 2) = 1)*(3*n^2 - 5*n - 5)/32 + IF(MOD(n, 3) = 1)*2/9.
%F G.f.: -x^3*(2*x^9 +x^8 -8*x^7 -9*x^6 +3*x^5 +29*x^4 +24*x^3 +14*x^2 +3*x +1) / ((x -1)^7*(x +1)^3*(x^2 +x +1)). - _Colin Barker_, Mar 23 2014
%e There are a(4) = 6 non-equivalent ways to place 3 points on a triangular grid of side 4:
%e . X X X X X
%e . X . . . . . . . . . .
%e X . . X . X X . . X . . . X . . . .
%e . . X . . . . . . . X . . . . X . . . X X . . X
%o (PARI) concat(0, Vec(-x^3*(2*x^9 +x^8 -8*x^7 -9*x^6 +3*x^5 +29*x^4 +24*x^3 +14*x^2 +3*x +1)/((x -1)^7*(x +1)^3*(x^2 +x +1)) + O(x^100))) \\ _Colin Barker_, Mar 23 2014
%Y Cf. A239572, A239569, A032091 (2 points), A239574 (4 points), A239575 (5 points), A279446 (6 points).
%K nonn,easy
%O 2,3
%A _Heinrich Ludwig_, Mar 23 2014
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