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.

1, 4, 0, 9, 12, 0, 16, 40, 24, 0, 25, 84, 160, 24, 0, 36, 144, 408, 496, 0, 0, 49, 220, 768, 1764, 1208, 0, 0, 64, 312, 1240, 3768, 6712, 2240, 0, 0, 81, 420, 1824, 6508, 17280, 22672, 2984, 0, 0, 100, 544, 2520, 9984, 32520, 74072, 68272, 2384, 0, 0, 121, 684

Offset

1,2

Comments

Table starts

1 4 9 16 25 36 49 64 81 100

0 12 40 84 144 220 312 420 544 684

0 24 160 408 768 1240 1824 2520 3328 4248

0 24 496 1764 3768 6508 9984 14196 19144 24828

0 0 1208 6712 17280 32520 52432 77016 106272 140200

0 0 2240 22672 74072 156484 268048 408764 578632 777652

0 0 2984 68272 296360 722384 1335984 2129440 3102752 4255920

0 0 2384 183472 1110000 3193800 6481216 10899404 16418600 23038804

0 0 784 436984 3908376 13530576 30543072 54738536 85743256

0 0 0 905776 12956800 55056168 139775784

Links

R. H. Hardin, Table of n, a(n) for n = 1..145

Formula

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.

Example

Some n=3 solutions for 3 X 3:
  3 2 0   0 0 0   0 3 0   0 0 0   0 0 0   0 0 1   0 1 0
  1 0 0   1 0 0   0 2 0   1 2 0   2 3 0   0 2 0   2 0 0
  0 0 0   2 3 0   0 0 1   3 0 0   0 1 0   0 0 3   0 3 0

Crossrefs

Rows are A000290, A033586(n-1), A186862, A186863, A186864, A186865, A186866, A186867, A366829.

Sequence in context: A247127 A277896 A195056 * A187027 A187850 A186965

Adjacent sequences: A186858 A186859 A186860 * A186862 A186863 A186864

Keyword

nonn,tabl

Author

R. H. Hardin, Feb 27 2011

Status

approved