A278683 Number of non-equivalent ways to place 4 non-attacking ferses on an n X n board.

0, 0, 6, 92, 832, 4500, 18229, 58881, 163509, 401259, 898420, 1861146, 3625546, 6694982, 11829267, 20099815, 33036079, 52700901, 81916834, 124362664, 184907220, 269726216, 386776561, 545930397, 759628777, 1043027055, 1414873104, 1897655046, 2518755934, 3310591194

Offset

1,3

Comments

A fers is a leaper [1, 1].

Rotations and reflections of placements are not counted. If they are to be counted, see A201245.

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 - 30*n^6 + 48*n^5 + 328*n^4 - 1056*n^3 - 200*n^2 + 4176*n - 4032 + IF(MOD(n, 2) = 1, 14*n^4 - 48*n^3 - 38*n^2 + 336*n - 459))/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*(6 +68*x +470*x^2 +1360*x^3 +2419*x^4 +1909*x^5 +836*x^6 -232*x^7 -192*x^8 +30*x^9 +54*x^10 -9*x^12 +x^13) / ((1 -x)^9*(1 +x)^5). - Colin Barker, Dec 10 2016

Example

There are 6 ways to place 4 non-attacking ferses on a 3 X 3 board rotations and reflections being ignored:
   XXX   XXX   X.X   X.X   XX.   XX.
   ...   ...   ...   ...   ...   ...
   ..X   .X.   X.X   XX.   XX.   .XX

Mathematica

Table[Boole[n > 2] (n^8 - 30 n^6 + 48 n^5 + 328 n^4 - 1056 n^3 - 200 n^2 + 4176 n - 4032 + Boole[OddQ@ n] (14 n^4 - 48 n^3 - 38 n^2 + 336 n - 459))/192, {n, 30}] (* Michael De Vlieger, Nov 30 2016 *)

Prog

(PARI) concat(vector(2), Vec(x^3*(6 +68*x +470*x^2 +1360*x^3 +2419*x^4 +1909*x^5 +836*x^6 -232*x^7 -192*x^8 +30*x^9 +54*x^10 -9*x^12 +x^13) / ((1 -x)^9*(1 +x)^5) + O(x^40))) \\ Colin Barker, Dec 10 2016

Crossrefs

Cf. A201245, A232567 (2 ferses), A278682 (3 ferses), A278684 (5 ferses), A278685 (6 ferses), A278686 (7 ferses), A278687, A278688.

Sequence in context: A182263 A360826 A171910 * A280214 A113266 A009607

Adjacent sequences: A278680 A278681 A278682 * A278684 A278685 A278686

Keyword

nonn,easy

Author

Heinrich Ludwig, Nov 26 2016

Status

approved