OFFSET
1,4
COMMENTS
Rotations and reflections of placements are not counted. If they are to be counted, see A061997.
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)
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
EXAMPLE
There is 1 way to place 4 non-attacking kings on a 3 X 3 board:
K.K
...
K.K
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Heinrich Ludwig, Dec 07 2016
STATUS
approved