OFFSET
1,2
EXAMPLE
On an n X n chessboard, a king in a corner is allowed to have n moves. For n=2, let's name the cells A1,A2,B1,B2 with the king at A1. Two moves, without repeating cells, can be done in the following 6 different ways: {A1-A2-B1, A1-A2-B2, A1-B1-A2, A1-B1-B2, A1-B2-A2, A1-B2-B1}. So a(2)=6.
MAPLE
pathCount := proc (N)
local g1, g2, nStep, gg, nCells, nPrev, i1, i2, j1, j2, i, j, nNext;
nCells := N^2; g1 := [[1]];
if N = 1 then return nops(g1) fi; #forced value for N=0
for nStep to N do
g2 := [];
for gg in g1 do
nPrev := gg[-1];
i1 := `if`(floor((nPrev-1)/N) = 0, 0, -N);
i2 := `if`(floor((nPrev-1)/N) = N-1, 0, N);
j1 := `if`(`mod`(nPrev-1, N) = 0, 0, -1);
j2 := `if`(`mod`(nPrev-1, N) = N-1, 0, 1);
for i from i1 by N to i2 do
for j from j1 to j2 do
if i = 0 and j = 0 then next fi;
nNext := nPrev+i+j;
if nNext < 0 or nCells < nNext or (nNext in gg) then next fi;
g2 := [op(g2), [op(gg), nNext]]
end do
end do
end do;
g1 := g2
end do;
return nops(g1);
end proc:
[seq(pathCount(n), n = 1 .. 6)];
CROSSREFS
KEYWORD
nonn,walk,more
AUTHOR
César Eliud Lozada, Apr 30 2016
EXTENSIONS
a(9)-a(16) from Alois P. Heinz, May 01 2016
a(17)-a(18) from Bert Dobbelaere, Jan 08 2019
STATUS
approved