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Number of 6-step king's tours on an n X n board summed over all starting positions.
1

%I #22 Dec 18 2023 11:29:45

%S 0,0,2240,22672,74072,156484,268048,408764,578632,777652,1005824,

%T 1263148,1549624,1865252,2210032,2583964,2987048,3419284,3880672,

%U 4371212,4890904,5439748,6017744,6624892,7261192,7926644,8621248,9345004,10097912,10879972,11691184

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

%C From _J. Volkmar Schmidt_, Oct 18 2023: (Start)

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

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

%C [0 0 0 0 0 2]

%C [0 0 96 576 448 124]

%C [0 96 1856 2272 1064 224]

%C [0 576 2272 1552 560 104]

%C [0 448 1064 560 168 28]

%C [2 124 224 104 28 4]

%C (End)

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, -3, 1).

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

%e Some solutions for 3X3

%e ..0..2..0....0..4..0....0..4..3....0..0..0....0..2..1....3..2..0....0..4..5

%e ..1..3..4....2..3..5....6..5..2....1..2..6....0..4..3....4..0..1....3..0..6

%e ..6..5..0....1..6..0....0..0..1....3..4..5....0..5..6....6..5..0....1..2..0

%Y Row 6 of A186861.

%K nonn

%O 1,3

%A _R. H. Hardin_, Feb 27 2011