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%I #29 Sep 08 2022 08:46:07
%S 0,1,114,1137,6100,23265,71211,186739,436437,932850,1856305,3483546,
%T 6224439,10668112,17640000,28271370,44083006,67084839,99893412,
%U 145869175,209275710,295463091,411077689,564300837,765118875,1025627200,1360371051,1786725864,2325320137
%N Number of ways to place 4 points on a triangular grid of side n so that no two of them are adjacent.
%C Rotations and reflections of placements are counted. If they are to be ignored see A239574.
%H Vincenzo Librandi, <a href="/A239570/b239570.txt">Table of n, a(n) for n = 3..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1)
%F a(n) = (n-2)*(n-3)*(n^6+9*n^5-39*n^4-353*n^3+950*n^2+4040*n-11904)/384.
%F G.f.: x^4*(38*x^6-156*x^5+153*x^4+113*x^3-147*x^2-105*x-1) / (x-1)^9. - _Colin Barker_, Mar 22 2014
%t CoefficientList[Series[x (38 x^6 - 156 x^5 + 153 x^4 + 113 x^3 - 147 x^2 - 105 x - 1)/(x - 1)^9, {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 23 2014 *)
%o (PARI) concat(0, Vec(x^4*(38*x^6-156*x^5+153*x^4+113*x^3-147*x^2-105*x-1)/(x-1)^9 + O(x^100))) \\ _Colin Barker_, Mar 22 2014
%o (Magma) [(n^2-5*n+6)*(n^6+9*n^5-39*n^4-353*n^3+950*n^2 +4040*n-11904)/384: n in [3..40]]: // _Vincenzo Librandi_, Mar 23 2014
%Y Cf. A239567, A239574, A239568 (2 points), A239569 (3 points), A239571 (5 points), A282998 (6 points).
%K nonn,easy
%O 3,3
%A _Heinrich Ludwig_, Mar 22 2014