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%I #18 Dec 10 2016 11:58:48
%S 0,0,6,92,832,4500,18229,58881,163509,401259,898420,1861146,3625546,
%T 6694982,11829267,20099815,33036079,52700901,81916834,124362664,
%U 184907220,269726216,386776561,545930397,759628777,1043027055,1414873104,1897655046,2518755934,3310591194
%N Number of non-equivalent ways to place 4 non-attacking ferses on an n X n board.
%C A fers is a leaper [1, 1].
%C Rotations and reflections of placements are not counted. If they are to be counted, see A201245.
%H Heinrich Ludwig, <a href="/A278683/b278683.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1,-16,19,20,-45,0,45,-20,-19,16,1,-4,1).
%F 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.
%F 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.
%F 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
%e There are 6 ways to place 4 non-attacking ferses on a 3 X 3 board rotations and reflections being ignored:
%e XXX XXX X.X X.X XX. XX.
%e ... ... ... ... ... ...
%e ..X .X. X.X XX. XX. .XX
%t 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 *)
%o (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
%Y Cf. A201245, A232567 (2 ferses), A278682 (3 ferses), A278684 (5 ferses), A278685 (6 ferses), A278686 (7 ferses), A278687, A278688.
%K nonn,easy
%O 1,3
%A _Heinrich Ludwig_, Nov 26 2016
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