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A282998 Number of ways to place 6 points on a triangular grid of side n so that no two of them are adjacent.

%I A282998 #20 Apr 11 2017 04:36:13
%S A282998 0,0,1,353,12231,153194,1124820,5893221,24425212,85152341,259805430,
%T A282998 712840480,1793423456,4197531636,9240962666,19301854131,38514786780,
%U A282998 73828909906,136581190475,244784427831,426389859697,723857976770,1200460734396,1948846090829,3102524331336
%N A282998 Number of ways to place 6 points on a triangular grid of side n so that no two of them are adjacent.
%C A282998 Rotations and reflections of placements are counted. If they are to be ignored, see A279446.
%H A282998 Heinrich Ludwig, <a href="/A282998/b282998.txt">Table of n, a(n) for n = 3..1000</a>
%H A282998 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).
%F A282998 a(n) = (n^12 + 6*n^11 - 195*n^10 - 670*n^9 + 17455*n^8 + 13426*n^7 - 836249*n^6 + 1252990*n^5 + 19599884*n^4 - 68542552*n^3 - 131400416*n^2 + 974223360*n - 1308856320)/46080 for n>=4.
%F A282998 G.f.: x^5*(1 + 340*x + 7720*x^2 + 21439*x^3 - 12927*x^4 - 27265*x^5 + 28385*x^6 - 6252*x^7 - 116*x^8 - 2365*x^9 + 1787*x^10 - 352*x^11) / (1 - x)^13. - _Colin Barker_, Feb 26 2017
%e A282998 There is a(5) = 1 way to place 6 points on a triangular grid of side n = 5:
%e A282998         X
%e A282998        . .
%e A282998       X . X
%e A282998      . . . .
%e A282998     X . X . X
%p A282998 A282998:=n->(n^12 + 6*n^11 - 195*n^10 - 670*n^9 + 17455*n^8 + 13426*n^7 - 836249*n^6 + 1252990*n^5 + 19599884*n^4 - 68542552*n^3 - 131400416*n^2 + 974223360*n - 1308856320)/46080: 0,seq(A282998(n), n=4..30); # _Wesley Ivan Hurt_, Apr 10 2017
%t A282998 Drop[CoefficientList[Series[(x^5 * (1 + 340 * x + 7720 * x^2 + 21439 * x^3 - 12927 * x^4 - 27265 * x^5 + 28385 * x^6 - 6252 * x^7 - 116 * x^8 - 2365 * x^9 + 1787 * x^10 - 352 * x^11) / (1 - x)^13 ),{x,0,27}],x],3] (* _Indranil Ghosh_, Feb 26 2017, from the g.f. by _Colin Barker_ *)
%o A282998 (PARI) concat(vector(2), Vec(x^5*(1 + 340*x + 7720*x^2 + 21439*x^3 - 12927*x^4 - 27265*x^5 + 28385*x^6 - 6252*x^7 - 116*x^8 - 2365*x^9 + 1787*x^10 - 352*x^11) / (1 - x)^13 + O(x^30))) \\ _Colin Barker_, Feb 26 2017
%Y A282998 Cf. A279446, A239567, A239568 (2 points), A239569 (3 points), A239570 (4 points), A239571 (5 points).
%K A282998 nonn,easy
%O A282998 3,4
%A A282998 _Heinrich Ludwig_, Feb 26 2017