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A279115 Number of non-equivalent ways to place 6 non-attacking kings on an n X n board. 8
0, 0, 0, 0, 143, 7855, 153311, 1505465, 9729830, 47235703, 186615092, 630338668, 1882894541, 5092130575, 12686490993, 29498296651, 64664954532, 134715649055, 268438970166, 514318521438, 951646716171, 1706721390223, 2976056379875, 5058962536429, 8402677784738, 13663807273607 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
Rotations and reflections of placements are not counted. If they are to be counted, see A172158.
LINKS
Index entries for linear recurrences with constant coefficients, signature (6,-8,-22,69,-8,-176,168,182,-364,0,364,-182,-168,176,8,-69,22,8,-6,1).
FORMULA
a(n) = (n^12 - 135*n^10 + 180*n^9 + 7465*n^8 - 18840*n^7 - 202468*n^6 + 749880*n^5 + 2446764*n^4 - 13439400*n^3 - 3570352*n^2 + 89413920*n - 107694720 + IF(MOD(n, 2) = 1, 122*n^6 - 1020*n^5 + 1955*n^4 + 840*n^3 + 5753*n^2 - 42840*n + 132975))/5760 for n>=5.
a(n) = 6*a(n-1) - 8*a(n-2) - 22*a(n-3) + 69*a(n-4) - 8*a(n-5) - 176*a(n-6) + 168*a(n-7) + 182*a(n-8) - 364*a(n-9) + 364*a(n-11) - 182*a(n-12) - 168*a(n-13) + 176*a(n-14) + 8*a(n-15) - 69*a(n-16) + 22*a(n-17) + 8*a(n-18) - 6*a(n-19) + a(n-20) for n>=25.
G.f.: x^5*(143 +6997*x +107325*x^2 +651585*x^3 +2086471*x^4 +3732434*x^5 +3669293*x^6 +1297859*x^7 -708745*x^8 -592136*x^9 +247421*x^10 +258649*x^11 -53671*x^12 -77714*x^13 +4451*x^14 +14969*x^15 +1018*x^16 -1741*x^17 -234*x^18 +106*x^19) / ((1 -x)^13*(1 +x)^7). - Colin Barker, Dec 09 2016
EXAMPLE
There are 143 non-equivalent ways to place 6 non-attacking kings on a 5 X 5 board, e.g., this one:
K...K
.....
K...K
.....
K...K
PROG
(PARI) concat(vector(4), Vec(x^5*(143 +6997*x +107325*x^2 +651585*x^3 +2086471*x^4 +3732434*x^5 +3669293*x^6 +1297859*x^7 -708745*x^8 -592136*x^9 +247421*x^10 +258649*x^11 -53671*x^12 -77714*x^13 +4451*x^14 +14969*x^15 +1018*x^16 -1741*x^17 -234*x^18 +106*x^19) / ((1 -x)^13*(1 +x)^7) + O(x^30))) \\ Colin Barker, Dec 09 2016
CROSSREFS
Cf. A172158, A279111 (2 kings), A279112 (3 kings), A279113 (4 kings), A279114 (5 kings), A279116 (7 kings), A279117, A236679.
Sequence in context: A159054 A135946 A217534 * A199039 A199235 A029555
KEYWORD
nonn,easy
AUTHOR
Heinrich Ludwig, Dec 09 2016
STATUS
approved

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Last modified February 26 02:39 EST 2024. Contains 370335 sequences. (Running on oeis4.)