A279113 Number of non-equivalent ways to place 4 non-attacking kings on an n X n board.

0, 0, 1, 14, 277, 2154, 10855, 39926, 120961, 315150, 737089, 1577406, 3150841, 5934034, 10651567, 18332614, 30452605, 49011606, 76753681, 117268590, 175315789, 256949306, 369978631, 524114454, 731604457, 1007394974, 1369985905, 1841600286, 2449309201, 3225197730

Offset: 1

Author

Heinrich Ludwig, Dec 07 2016

Keywords

• nonn
• easy

Comments

Rotations and reflections of placements are not counted. If they are to be counted, see A061997.

Examples

			There is 1 way to place 4 non-attacking kings on a 3 X 3 board:
   K.K
   ...
   K.K
		

Links

• Heinrich Ludwig, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-1,-16,19,20,-45,0,45,-20,-19,16,1,-4,1)

Crossrefs

Cf. A061997, A279111 (2 kings), A279112 (3 kings), A279114 (5 kings), A279115 (6 kings), A279116 (7 kings), A279117, A236679.

Programs

Mathematica

Table[Boole[n > 2] (n^8 - 54 n^6 + 72 n^5 + 1024 n^4 - 2640 n^3 - 4928 n^2 + 21888 n - 17280 + Boole[OddQ@ n] (14 n^4 - 72 n^3 + 154 n^2 - 240 n - 51))/192, {n, 30}] (* or *)
Rest@ CoefficientList[Series[x^3*(1 + 10 x + 222 x^2 + 1076 x^3 + 2721 x^4 + 2806 x^5 + 1078 x^6 - 924 x^7 - 639 x^8 + 202 x^9 + 236 x^10 - 40 x^11 - 35 x^12 + 6 x^13)/((1 - x)^9*(1 + x)^5), {x, 0, 30}], x] (* Michael De Vlieger, Dec 08 2016 *)

PARI

concat(vector(2), Vec(x^3*(1 +10*x +222*x^2 +1076*x^3 +2721*x^4 +2806*x^5 +1078*x^6 -924*x^7 -639*x^8 +202*x^9 +236*x^10 -40*x^11 -35*x^12 +6*x^13) / ((1 -x)^9*(1 +x)^5) + O(x^40))) \\ Colin Barker, Dec 08 2016

Formula

a(n) = (n^8 - 54*n^6 + 72*n^5 + 1024*n^4 - 2640*n^3 - 4928*n^2 + 21888*n - 17280 + IF(MOD(n, 2) = 1, 14*n^4 - 72*n^3 + 154*n^2 - 240*n - 51))/192 for n>=3.

a(n) = 4*a(n-1) - a(n-2) - 16*a(n-3) + 19*a(n-4) + 20*a(n-5) - 45*a(n-6) + 45*a(n-8) - 20*a(n-9) - 19*a(n-10) + 16*a(n-11) + a(n-12) - 4*a(n-13) + a(n-14) for n>=17.

G.f.: x^3*(1 +10*x +222*x^2 +1076*x^3 +2721*x^4 +2806*x^5 +1078*x^6 -924*x^7 -639*x^8 +202*x^9 +236*x^10 -40*x^11 -35*x^12 +6*x^13) / ((1 -x)^9*(1 +x)^5). - Colin Barker, Dec 08 2016