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 A201244.
LINKS
Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Wikipedia, Fairy chess piece
Index entries for linear recurrences with constant coefficients, signature (3,1,-11,6,14,-14,-6,11,-1,-3,1).
FORMULA
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.
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).
From Colin Barker, Dec 07 2016: (Start)
a(n) = (n^6 - 15*n^4 + 32*n^3 + 14*n^2 - 116*n + 96)/48 for n even.
a(n) = (n^6 - 15*n^4 + 40*n^3 + 5*n^2 - 136*n + 105)/48 for n odd.
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).
(End)
EXAMPLE
There are 7 ways to place 3 non-attacking ferses "X" on a 3 X 3 board, rotations and reflections being ignored
XXX XX. X.X ... X.. X.. X..
... ... ... XXX X.X ... ...
... ..X .X. ... ... XX. X.X
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Heinrich Ludwig, Nov 26 2016
STATUS
approved