A187172 T(n,k) is the number of n-step left-handed knight's tours (moves only out two, left one) on a k X k board summed over all starting positions.

1, 4, 0, 9, 0, 0, 16, 8, 0, 0, 25, 24, 0, 0, 0, 36, 48, 16, 0, 0, 0, 49, 80, 60, 8, 0, 0, 0, 64, 120, 128, 48, 0, 0, 0, 0, 81, 168, 220, 176, 16, 0, 0, 0, 0, 100, 224, 336, 384, 136, 0, 0, 0, 0, 0, 121, 288, 476, 664, 456, 88, 0, 0, 0, 0, 0, 144, 360, 640, 1016, 1024, 496, 16, 0, 0, 0, 0, 0

Offset

1,2

Comments

Table starts

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256

0 0 8 24 48 80 120 168 224 288 360 440 528 624 728 840

0 0 0 16 60 128 220 336 476 640 828 1040 1276 1536 1820 2128

0 0 0 8 48 176 384 664 1016 1440 1936 2504 3144 3856 4640 5496

0 0 0 0 16 136 456 1024 1804 2784 3964 5344 6924 8704 10684 12864

0 0 0 0 0 88 496 1440 3064 5344 8208 11640 15640 20208 25344 31048

0 0 0 0 0 16 368 1600 4284 8760 15104 23144 32764 43944 56684 70984

0 0 0 0 0 0 280 1784 5944 14072 27104 45288 68400 96048 128064 164424

0 0 0 0 0 0 88 1440 6828 19840 43668 80624 131576 196192 273592 363080

0 0 0 0 0 0 8 1088 7896 27984 70344 142816 250728 396808 580696 800584

Links

R. H. Hardin, Table of n, a(n) for n = 1..310

Formula

Empirical:

T(1,k) = k^2;

T(2,k) = 4*k^2 - 12*k + 8;

T(3,k) = 12*k^2 - 64*k + 80 for k > 3;

T(4,k) = 36*k^2 - 260*k + 440 for k > 5;

T(5,k) = 100*k^2 - 920*k + 1984 for k > 7;

T(6,k) = 284*k^2 - 3100*k + 7944 for k > 9;

T(7,k) = 780*k^2 - 9880*k + 29384 for k > 11;

T(8,k) = 2172*k^2 - 30972*k + 103944 for k > 13.

Example

One of 98568 n=51 solutions for 16 X 16:
   0  1  0  0  0  0  4  0  0  0  0  7  0  0  0  0
   0  0  0  0  3  0  0  0  0  6  0  0  0  0  9  0
   0  0  2  0  0  0  0  5  0  0  0  0  8  0  0  0
  51  0  0  0  0 16  0  0  0  0 13  0  0  0  0 10
   0  0  0 17  0  0  0  0 14  0  0  0  0 11  0  0
   0 50  0  0  0  0 15  0  0  0  0 12  0  0  0  0
   0  0  0  0 18  0  0  0  0 21  0  0  0  0 24  0
   0  0 49  0  0  0  0 20  0  0  0  0 23  0  0  0
  48  0  0  0  0 19  0  0  0  0 22  0  0  0  0 25
   0  0  0 46  0  0  0  0 35  0  0  0  0 26  0  0
   0 47  0  0  0  0 36  0  0  0  0 27  0  0  0  0
   0  0  0  0 45  0  0  0  0 34  0  0  0  0 29  0
   0  0 44  0  0  0  0 37  0  0  0  0 28  0  0  0
  43  0  0  0  0 40  0  0  0  0 33  0  0  0  0 30
   0  0  0 41  0  0  0  0 38  0  0  0  0 31  0  0
   0 42  0  0  0  0 39  0  0  0  0 32  0  0  0  0

Crossrefs

Row 2 is A033996(n-2).

Sequence in context: A260491 A013440 A013496 * A296570 A241667 A259258

Adjacent sequences: A187169 A187170 A187171 * A187173 A187174 A187175

Keyword

nonn,tabl

Author

R. H. Hardin, Mar 06 2011

Status

approved