The a(n) represent the number of n-move paths of a chess king on a 3 X 3 board that end or start in a given corner square m (m = 1, 3, 7, 9). To determine the a(n) we can either sum the components of the column vector A^n[k,m], with A the adjacency matrix of the king's graph, or we can sum the components of the row vector A^n[m,k], see the Maple program.
Inverse binomial transform of
A079291 (without the leading 0).
(End)
The row n=3 of an array counting king walks on an n X n board with k steps, starting from a corner:
1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, ...;
1, 3, 18, 80, 400, 1904, 9248, 44544, 215296, 1039104, 5018112, ...;
1, 3, 18, 105, 615, 3600, 21075, 123375, 722250, 4228125, 24751875, ...;
1, 3, 18, 105, 684, 4359, 28278, 182349, 1179792, 7622667, 49283802, ...;
1, 3, 18, 105, 684, 4550, 30807, 209867, 1434279, 9815190, 67209723, ...;
1, 3, 18, 105, 684, 4550, 31340, 218056, 1533712, 10829360, 76720288, ...;
1, 3, 18, 105, 684, 4550, 31340, 219555, 1559835, 11177190, 80573373, ...;
1, 3, 18, 105, 684, 4550, 31340, 219555, 1564080, 11259785, 81765550, ...;
1, 3, 18, 105, 684, 4550, 31340, 219555, 1564080, 11271876, 82025163, ...;
1, 3, 18, 105, 684, 4550, 31340, 219555, 1564080, 11271876, 82059768, ...;
1, 3, 18, 105, 684, 4550, 31340, 219555, 1564080, 11271876, 82059768, ...;
The partial sums along the rows are documented in
A123109 (king walks with between 1 and k steps). (End)
