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%I #10 Dec 21 2016 12:18:02
%S 0,0,0,0,143,7855,153311,1505465,9729830,47235703,186615092,630338668,
%T 1882894541,5092130575,12686490993,29498296651,64664954532,
%U 134715649055,268438970166,514318521438,951646716171,1706721390223,2976056379875,5058962536429,8402677784738,13663807273607
%N Number of non-equivalent ways to place 6 non-attacking kings on an n X n board.
%C Rotations and reflections of placements are not counted. If they are to be counted, see A172158.
%H Heinrich Ludwig, <a href="/A279115/b279115.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (6,-8,-22,69,-8,-176,168,182,-364,0,364,-182,-168,176,8,-69,22,8,-6,1).
%F a(n) = (n^12 - 135*n^10 + 180*n^9 + 7465*n^8 - 18840*n^7 - 202468*n^6 + 749880*n^5 + 2446764*n^4 - 13439400*n^3 - 3570352*n^2 + 89413920*n - 107694720 + IF(MOD(n, 2) = 1, 122*n^6 - 1020*n^5 + 1955*n^4 + 840*n^3 + 5753*n^2 - 42840*n + 132975))/5760 for n>=5.
%F a(n) = 6*a(n-1) - 8*a(n-2) - 22*a(n-3) + 69*a(n-4) - 8*a(n-5) - 176*a(n-6) + 168*a(n-7) + 182*a(n-8) - 364*a(n-9) + 364*a(n-11) - 182*a(n-12) - 168*a(n-13) + 176*a(n-14) + 8*a(n-15) - 69*a(n-16) + 22*a(n-17) + 8*a(n-18) - 6*a(n-19) + a(n-20) for n>=25.
%F G.f.: x^5*(143 +6997*x +107325*x^2 +651585*x^3 +2086471*x^4 +3732434*x^5 +3669293*x^6 +1297859*x^7 -708745*x^8 -592136*x^9 +247421*x^10 +258649*x^11 -53671*x^12 -77714*x^13 +4451*x^14 +14969*x^15 +1018*x^16 -1741*x^17 -234*x^18 +106*x^19) / ((1 -x)^13*(1 +x)^7). - _Colin Barker_, Dec 09 2016
%e There are 143 non-equivalent ways to place 6 non-attacking kings on a 5 X 5 board, e.g., this one:
%e K...K
%e .....
%e K...K
%e .....
%e K...K
%o (PARI) concat(vector(4), Vec(x^5*(143 +6997*x +107325*x^2 +651585*x^3 +2086471*x^4 +3732434*x^5 +3669293*x^6 +1297859*x^7 -708745*x^8 -592136*x^9 +247421*x^10 +258649*x^11 -53671*x^12 -77714*x^13 +4451*x^14 +14969*x^15 +1018*x^16 -1741*x^17 -234*x^18 +106*x^19) / ((1 -x)^13*(1 +x)^7) + O(x^30))) \\ _Colin Barker_, Dec 09 2016
%Y Cf. A172158, A279111 (2 kings), A279112 (3 kings), A279113 (4 kings), A279114 (5 kings), A279116 (7 kings), A279117, A236679.
%K nonn,easy
%O 1,5
%A _Heinrich Ludwig_, Dec 09 2016
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