Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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