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A186867
Number of 8-step king's tours on an n X n board summed over all starting positions.
2
0, 0, 2384, 183472, 1110000, 3193800, 6481216, 10899404, 16418600, 23038804, 30760016, 39582236, 49505464, 60529700, 72654944, 85881196, 100208456, 115636724, 132166000, 149796284, 168527576, 188359876, 209293184, 231327500, 254462824, 278699156
OFFSET
1,3
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 8-step walks is:
[0 0 0 0 0 0 0 2]
[0 0 0 416 3264 4224 2304 508]
[0 0 2384 26004 38120 26164 10080 1764]
[0 416 26004 67424 53320 26480 8460 1328]
[0 3264 38120 53320 32032 13428 3816 560]
[0 4224 26164 26480 13428 4952 1260 172]
[0 2304 10080 8460 3816 1260 288 36]
[2 508 1764 1328 560 172 36 4]
(End)
FORMULA
Empirical: a(n) = 550504*n^2 - 3839372*n + 6382124 for n > 6.
EXAMPLE
Some solutions for 4 X 4:
0 7 6 0 2 1 0 8 0 0 1 0 0 0 6 8 3 4 5 0
8 0 5 1 4 3 7 0 0 0 3 2 0 0 7 5 2 0 6 0
0 4 3 2 0 5 6 0 0 7 5 4 2 1 4 0 1 0 7 8
0 0 0 0 0 0 0 0 0 8 6 0 0 3 0 0 0 0 0 0
CROSSREFS
Row 8 of A186861.
Sequence in context: A251649 A218849 A254562 * A210078 A172653 A172677
KEYWORD
nonn
AUTHOR
R. H. Hardin, Feb 27 2011
EXTENSIONS
a(12)-a(26) from J. Volkmar Schmidt, Aug 27 2023
STATUS
approved