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Number of non-equivalent ways to place 6 non-attacking ferses on an n X n board.
6

%I #13 Dec 10 2016 10:48:47

%S 0,0,1,76,3773,66201,651193,4318451,21754341,89267490,312974387,

%T 968069337,2704548145,6942663519,16594368633,37311795887,79570707969,

%U 162013125016,316669793867,596873304925,1089009784181,1929545889877,3329316638249,5607471933963,9238336533613

%N Number of non-equivalent ways to place 6 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 A201247.

%H Heinrich Ludwig, <a href="/A278685/b278685.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_20">Index entries for linear recurrences with constant coefficients</a>, signature (6,-8,-22,69,-8,-176,168,182,-364,0,364,-182,-168,176,8,-69,22,8,-6,1).

%F a(n) = n^12 - 75*n^10 + 120*n^9 + 2305*n^8 - 6960*n^7 - 32008*n^6 + 152880*n^5 + 138204*n^4 - 1543560*n^3 + 1178528*n^2 + 5238720*n - 7977600 + IF(MOD(n, 2) = 1, 122*n^6 - 600*n^5 - 1645*n^4 + 14520*n^3 - 19447*n^2 - 30480*n + 81855)/5760 for n>=5.

%F 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.

%F G.f.: x^3*(1 +70*x +3325*x^2 +44193*x^3 +285774*x^4 +1018671*x^5 +2250048*x^6 +3090821*x^7 +2658486*x^8 +1198906*x^9 +139256*x^10 -84845*x^11 +22114*x^12 +28024*x^13 -6172*x^14 -5377*x^15 +485*x^16 +592*x^17 +199*x^18 -56*x^19 -44*x^20 +9*x^21) / ((1 -x)^13*(1 +x)^7). - _Colin Barker_, Dec 10 2016

%e There is 1 way to place 6 non-attacking ferses on a 3 X 3 board, rotations and reflections being ignored:

%e XXX

%e ...

%e XXX

%o (PARI) concat(vector(2), Vec(x^3*(1 +70*x +3325*x^2 +44193*x^3 +285774*x^4 +1018671*x^5 +2250048*x^6 +3090821*x^7 +2658486*x^8 +1198906*x^9 +139256*x^10 -84845*x^11 +22114*x^12 +28024*x^13 -6172*x^14 -5377*x^15 +485*x^16 +592*x^17 +199*x^18 -56*x^19 -44*x^20 +9*x^21) / ((1 -x)^13*(1 +x)^7) + O(x^30))) \\ _Colin Barker_, Dec 10 2016

%Y Cf. A201247, A232567 (2 ferses), A278682 (3 ferses), A278683 (4 ferses), A278684 (5 ferses), A278686 (7 ferses), A278687, A278688.

%K nonn,easy

%O 1,4

%A _Heinrich Ludwig_, Nov 26 2016