[seqfan] Rook path generates A006189 Number of paths on square lattice?

[David Scambler]

The description of A006189 is vague, but a rook generates the same 8 terms given.

Count of self-avoiding rook paths on a 3xn board from (0,0) to (0,n) generates 1,3,11,38,126..., apparently = A006189(n+2), i.e. offset 1 rather than 3.

3 x 1 a(1) = 1
  1  0  0

3 x 2 a(2) = 3
  1  0  0 :   1  2  0 :   1  2  3
  2  0  0 :   4  3  0 :   6  5  4


3 x 3 a(3) = 11
  1  0  0 :   1  0  0 :   1  4  5 :   1  0  0 :   1  2  0 :   1  2  0
  2  0  0 :   2  3  0 :   2  3  6 :   2  3  4 :   0  3  0 :   0  3  4
  3  0  0 :   5  4  0 :   9  8  7 :   7  6  5 :   5  4  0 :   7  6  5

  1  2  0 :   1  2  3 :   1  2  3 :   1  2  3 :   1  2  3
  4  3  0 :   8  7  4 :   0  0  4 :   0  5  4 :   6  5  4
  5  0  0 :   9  6  5 :   7  6  5 :   7  6  0 :   7  0  0


If anyone has the reference H. L. Abbott and D. Hanson, A lattice path problem, Ars Combin., 6 (1978), 163-178 I would be interested to know what the original motivation for this sequence was, and to determine whether, in fact, this is the same sequence as that generated by the rook paths.

Thanks, dave