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%I #26 Dec 07 2016 18:30:22
%S 0,0,7,45,225,709,1974,4524,9614,18382,33425,56895,93447,146715,
%T 224280,331814,480844,679724,945099,1288737,1733725,2296065,3006762,
%U 3886960,4977210,6304794,7921589,9862099,12191459,14952567,18225900,22064010,26564952,31792280
%N Number of non-equivalent ways to place 3 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 A201244.
%H Heinrich Ludwig, <a href="/A278682/b278682.txt">Table of n, a(n) for n = 1..1000</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fairy_chess_piece">Fairy chess piece</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-11,6,14,-14,-6,11,-1,-3,1).
%F a(n) = ((n^6 - 15*n^4 + 32*n^3 + 14*n^2 - 116*n + 96) + IF(MOD(n, 2) = 1, 8*n^3 - 9*n^2 - 20*n + 9))/48.
%F a(n) = 3*a(n-1) + a(n-2) - 11*a(n-3) + 6*a(n-4) + 14*a(n-5) - 14*a(n-6) - 6*a(n-7) + 11*a(n-8) - a(n-9) - 3*a(n-10) + a(n-11).
%F From _Colin Barker_, Dec 07 2016: (Start)
%F a(n) = (n^6 - 15*n^4 + 32*n^3 + 14*n^2 - 116*n + 96)/48 for n even.
%F a(n) = (n^6 - 15*n^4 + 40*n^3 + 5*n^2 - 136*n + 105)/48 for n odd.
%F G.f.: x^3*(7 + 24*x + 83*x^2 + 66*x^3 + 75*x^4 - 15*x^6 - 2*x^7 + 2*x^8) / ((1 - x)^7*(1 + x)^4).
%F (End)
%e There are 7 ways to place 3 non-attacking ferses "X" on a 3 X 3 board, rotations and reflections being ignored
%e XXX XX. X.X ... X.. X.. X..
%e ... ... ... XXX X.X ... ...
%e ... ..X .X. ... ... XX. X.X
%t Table[Boole[n > 2] ((n^6 - 15 n^4 + 32 n^3 + 14 n^2 - 116 n + 96) + Boole[OddQ@ n] (8 n^3 - 9 n^2 - 20 n + 9))/48, {n, 34}] (* _Michael De Vlieger_, Nov 30 2016 *)
%o (PARI) concat(vector(2), Vec(x^3*(7 + 24*x + 83*x^2 + 66*x^3 + 75*x^4 - 15*x^6 - 2*x^7 + 2*x^8) / ((1 - x)^7*(1 + x)^4) + O(x^40))) \\ _Colin Barker_, Dec 07 2016
%Y Cf. A201244, A232567 (2 ferses), A278683 (4 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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