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A186861 Array read by antidiagonals: T(n,k) is the number of n-step king's tours on a k X k board summed over all starting positions.

%I A186861 #26 Nov 29 2024 17:33:25
%S A186861 1,4,0,9,12,0,16,40,24,0,25,84,160,24,0,36,144,408,496,0,0,49,220,768,
%T A186861 1764,1208,0,0,64,312,1240,3768,6712,2240,0,0,81,420,1824,6508,17280,
%U A186861 22672,2984,0,0,100,544,2520,9984,32520,74072,68272,2384,0,0,121,684
%N A186861 Array read by antidiagonals: T(n,k) is the number of n-step king's tours on a k X k board summed over all starting positions.
%H A186861 R. H. Hardin, <a href="/A186861/b186861.txt">Table of n, a(n) for n = 1..145</a>
%F A186861 Empirical, for all rows: a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3,3,3,5,6,7,8,9 respectively for row=1..8.
%e A186861 Table starts:
%e A186861   1  4    9     16       25       36        49       64       81      100
%e A186861   0 12   40     84      144      220       312      420      544      684
%e A186861   0 24  160    408      768     1240      1824     2520     3328     4248
%e A186861   0 24  496   1764     3768     6508      9984    14196    19144    24828
%e A186861   0  0 1208   6712    17280    32520     52432    77016   106272   140200
%e A186861   0  0 2240  22672    74072   156484    268048   408764   578632   777652
%e A186861   0  0 2984  68272   296360   722384   1335984  2129440  3102752  4255920
%e A186861   0  0 2384 183472  1110000  3193800   6481216 10899404 16418600 23038804
%e A186861   0  0  784 436984  3908376 13530576  30543072 54738536 85743256
%e A186861   0  0    0 905776 12956800 55056168 139775784
%e A186861 Some n=3 solutions for 3 X 3:
%e A186861   3 2 0   0 0 0   0 3 0   0 0 0   0 0 0   0 0 1   0 1 0
%e A186861   1 0 0   1 0 0   0 2 0   1 2 0   2 3 0   0 2 0   2 0 0
%e A186861   0 0 0   2 3 0   0 0 1   3 0 0   0 1 0   0 0 3   0 3 0
%Y A186861 Rows are A000290, A033586(n-1), A186862, A186863, A186864, A186865, A186866, A186867, A366829.
%K A186861 nonn,tabl
%O A186861 1,2
%A A186861 _R. H. Hardin_, Feb 27 2011