[seqfan] Fairy chess sequences

[Charles Greathouse]

I've been trying to figure out A180140 recently and I thought I'd come here
for help. Its name is

Eight rooks and one berserker on a 3 X 3 chessboard. G.f.:
(1+x+x^2)/(1-3*x-5*x^2).

Of course the g.f. already defines the sequence, but I want to understand
the other part of the definition.

Now, for me at least, the name "Eight rooks and one berserker on a 3 X 3
chessboard" does not describe a sequence, but the first comment helps:

"a(n) represent the number of n-move routes of a fairy chess piece starting
in a given side square (m = 2, 4, 6 or 8) on a 3 X 3 chessboard. This fairy
chess piece behaves like a rook on the eight side and corner squares but on
the central square the rook goes berserk and turns into a berserker."

So we have a 3 X 3 board, which has 4 side squares, 4 corner squares, and
the center. We start on a side square and count the number of possibilities
for this piece. Now I don't know how a berserker moves, but the comment
tells me some of it: on the squares other than the center it acts like a
rook. Thus from the side it has four moves: one to another side, one to the
center, and two to corners. From a corner it has two moves to other corners
and two moves to sides. This should allow me to determine at least the
number of moves from the center.

a(0) = 1 = 1 side + 0 corner + 0 center.
a(1) = 4 = 1 side + 2 corner + 1 center.
a(2) = 18 = 5+x side + 6+y corner + 1+z center.

x + y + z + 12 = 18, thus the berzerker has 6 moves from the center. Wait,
6? There's some kind of symmetry-breaking here that looks like it's going
to defeat my groupings. But the sequence is defined only up to rotation:
"starting in a given side square", not "starting in the lower side square".

Can someone help here? There are also related sequences like A180146 with
similar definitions.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University