%I #16 Sep 08 2022 08:46:18

%S 0,0,0,0,273,5335,50021,291171,1263125,4434783,13355477,35672426,

%T 86686721,194886975,410820269,819819261,1561128613,2853802623,

%U 5033838173,8602315716,14291999441,23150803815,36654054741,56841404455,86496828245,129363299967,190419751685,276205278030

%N Number of non-equivalent ways to place 5 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 A061998.

%H Heinrich Ludwig, <a href="/A279114/b279114.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4,-20,40,16,-100,44,110,-110,-44,100,-16,-40,20,4,-5,1).

%F a(n) = (n^10 - 90*n^8 + 120*n^7 + 3115*n^6 - 7748*n^5 - 46050*n^4 + 173140*n^3 + 158584*n^2 - 1255952*n + 1252800 + (1/2-(-1)^n/2) * (52*n^5 - 305*n^4 + 180*n^3 + 320*n^2 + 3488*n - 375))/960 for n >= 4.

%F a(n) = 5*a(n-1) - 4*a(n-2) - 20*a(n-3) + 40*a(n-4) + 16*a(n-5) - 100*a(n-6) + 44*a(n-7) + 110*a(n-8) - 110*a(n-9) - 44*a(n-10) + 100*a(n-11) - 16*a(n-12) - 40*a(n-13) + 20*a(n-14) + 4*a(n-15) - 5*a(n-16) + a(n-17) for n >= 21.

%F G.f.: x^5*(273 +3970*x +24438*x^2 +67866*x^3 +103134*x^4 +66494*x^5 -1418*x^6 -29015*x^7 -4247*x^8 +10650*x^9 +2718*x^10 -2696*x^11 -672*x^12 +382*x^13 +62*x^14 -19*x^15) / ((1 -x)^11*(1 +x)^6). - _Colin Barker_, Dec 08 2016

%e There are 273 non-equivalent ways to place 5 non-attacking kings on a 5 X 5 board, e.g., this one:

%e K...K

%e .....

%e ..K..

%e .....

%e K...K

%p A279114:=n->(n^10 - 90*n^8 + 120*n^7 + 3115*n^6 - 7748*n^5 - 46050*n^4 + 173140*n^3 + 158584*n^2 - 1255952*n + 1252800 + (1/2-(-1)^n/2)*(52*n^5 - 305*n^4 + 180*n^3 + 320*n^2 + 3488*n - 375))/960: 0, 0, 0, seq(A279114(n), n=4..30); # _Wesley Ivan Hurt_, Dec 08 2016

%t Join[{0, 0, 0}, Table[(n^10 - 90*n^8 + 120*n^7 + 3115*n^6 - 7748*n^5 - 46050*n^4 + 173140*n^3 + 158584*n^2 - 1255952*n + 1252800 + (1/2 - (-1)^n/2)*(52*n^5 - 305*n^4 + 180*n^3 + 320*n^2 + 3488*n - 375))/960, {n, 4, 30}]] (* _Wesley Ivan Hurt_, Dec 08 2016 *)

%o (PARI) concat(vector(4), Vec(x^5*(273 +3970*x +24438*x^2 +67866*x^3 +103134*x^4 +66494*x^5 -1418*x^6 -29015*x^7 -4247*x^8 +10650*x^9 +2718*x^10 -2696*x^11 -672*x^12 +382*x^13 +62*x^14 -19*x^15) / ((1 -x)^11*(1 +x)^6) + O(x^30))) \\ _Colin Barker_, Dec 08 2016

%o (Magma) [0,0,0] cat [(n^10 - 90*n^8 + 120*n^7 + 3115*n^6 - 7748*n^5 - 46050*n^4 + 173140*n^3 + 158584*n^2 - 1255952*n + 1252800 + (1/2-(-1)^n/2)*(52*n^5 - 305*n^4 + 180*n^3 + 320*n^2 + 3488*n - 375))/960 : n in [4..30]]; // _Wesley Ivan Hurt_, Dec 08 2016

%Y Cf. A061998, A279111 (2 kings), A279112 (3 kings), A279113 (4 kings), A279115 (6 kings), A279116 (7 kings), A279117, A236679.

%K nonn,easy

%O 1,5

%A _Heinrich Ludwig_, Dec 08 2016