A279114 Number of non-equivalent ways to place 5 non-attacking kings on an n X n board.

0, 0, 0, 0, 273, 5335, 50021, 291171, 1263125, 4434783, 13355477, 35672426, 86686721, 194886975, 410820269, 819819261, 1561128613, 2853802623, 5033838173, 8602315716, 14291999441, 23150803815, 36654054741, 56841404455, 86496828245, 129363299967, 190419751685, 276205278030

Offset

1,5

Comments

Rotations and reflections of placements are not counted. If they are to be counted, see A061998.

Links

Heinrich Ludwig, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (5,-4,-20,40,16,-100,44,110,-110,-44,100,-16,-40,20,4,-5,1).

Formula

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.

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.

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

Example

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

Maple

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

Mathematica

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A214220 A029567 A028534 * A210270 A351766 A321035

Adjacent sequences: A279111 A279112 A279113 * A279115 A279116 A279117

Keyword

nonn,easy

Author

Heinrich Ludwig, Dec 08 2016

Status

approved