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A279113
Number of non-equivalent ways to place 4 non-attacking kings on an n X n board.
9
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
1,4
COMMENTS
Rotations and reflections of placements are not counted. If they are to be counted, see A061997.
LINKS
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
Cf. A061997, A279111 (2 kings), A279112 (3 kings), A279114 (5 kings), A279115 (6 kings), A279116 (7 kings), A279117, A236679.
Sequence in context: A211900 A215544 A205353 * A291099 A053101 A205746
KEYWORD
nonn,easy
AUTHOR
Heinrich Ludwig, Dec 07 2016
STATUS
approved