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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. 7
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified January 29 10:34 EST 2020. Contains 331337 sequences. (Running on oeis4.)