A186866 Number of 7-step king's tours on an n X n board summed over all starting positions.

0, 0, 2984, 68272, 296360, 722384, 1335984, 2129440, 3102752, 4255920, 5588944, 7101824, 8794560, 10667152, 12719600, 14951904, 17364064, 19956080, 22727952, 25679680, 28811264, 32122704, 35614000, 39285152, 43136160, 47167024

Offset: 1

Author

R. H. Hardin, Feb 27 2011

Keywords

• nonn

Comments

From J. Volkmar Schmidt, Oct 24 2023 (Start)

Proof of a(n) follows proof scheme from David A. Corneth for A186864.

Distribution matrix of surrounding rectangles for 7-step walks is:

[0 0 0 0 0 0 2]

[0 0 0 800 1600 1024 252]

[0 0 2984 9780 7904 3360 644]

[0 800 9780 12416 6636 2272 388]

[0 1600 7904 6636 2912 864 136]

[0 1024 3360 2272 864 224 32]

[2 252 644 388 136 32 4]

(End)

Examples

			Some solutions for 3X3
..0..2..1....0..2..0....6..0..0....1..7..6....1..2..3....7..2..1....6..5..0
..4..3..0....3..7..1....7..5..2....2..0..5....0..4..7....3..6..0....7..4..1
..5..6..7....4..5..6....4..3..1....3..4..0....0..6..5....5..4..0....0..2..3
		

Links

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

Crossrefs

Row 7 of A186861.

Formula

Empirical: a(n) = 89928*n^2 - 555464*n + 817760 for n>5.