A186865 Number of 6-step king's tours on an n X n board summed over all starting positions.

0, 0, 2240, 22672, 74072, 156484, 268048, 408764, 578632, 777652, 1005824, 1263148, 1549624, 1865252, 2210032, 2583964, 2987048, 3419284, 3880672, 4371212, 4890904, 5439748, 6017744, 6624892, 7261192, 7926644, 8621248, 9345004, 10097912, 10879972, 11691184

Offset: 1

Author

R. H. Hardin, Feb 27 2011

Keywords

• nonn

Comments

From J. Volkmar Schmidt, Oct 18 2023: (Start)

Proof of a(n) = 14576*n^2 - 77924*n + 99292 for n>4 follows proof scheme from David A. Corneth for A186864.

Distribution matrix of surrounding rectangles for 6-step walks is:

[0 0 0 0 0 2]

[0 0 96 576 448 124]

[0 96 1856 2272 1064 224]

[0 576 2272 1552 560 104]

[0 448 1064 560 168 28]

[2 124 224 104 28 4]

(End)

Examples

			Some solutions for 3X3
..0..2..0....0..4..0....0..4..3....0..0..0....0..2..1....3..2..0....0..4..5
..1..3..4....2..3..5....6..5..2....1..2..6....0..4..3....4..0..1....3..0..6
..6..5..0....1..6..0....0..0..1....3..4..5....0..5..6....6..5..0....1..2..0
		

Links

• Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Crossrefs

Row 6 of A186861.

Formula

Empirical: a(n) = 14576*n^2 - 77924*n + 99292 for n>4.