login
A186863
Number of 4-step king's tours on an n X n board summed over all starting positions.
2
0, 24, 496, 1764, 3768, 6508, 9984, 14196, 19144, 24828, 31248, 38404, 46296, 54924, 64288, 74388, 85224, 96796, 109104, 122148, 135928, 150444, 165696, 181684, 198408, 215868, 234064, 252996, 272664, 293068, 314208, 336084, 358696, 382044, 406128, 430948, 456504, 482796, 509824
OFFSET
1,2
COMMENTS
From J. Volkmar Schmidt, Oct 25 2023 (Start)
Proof of formula for a(n) follows proof scheme from David A. Corneth for A186864.
Distribution matrix of surrounding rectangles for 4-step walks is:
[0 0 0 2]
[0 24 80 28]
[0 80 80 20]
[2 28 20 4] (End)
FORMULA
a(n) = 368*n^2 - 1308*n + 1108 = 4*(92*(n-1)*(n-3) + 41*n + 1) for n > 2.
G.f.: 4*x^2*(6 + 106*x + 87*x^2 - 15*x^3)/(1-x)^3. - Colin Barker, Jan 22 2012
EXAMPLE
Some solutions for 3 X 3:
0 3 0 0 2 0 0 0 1 1 0 0 0 0 0 0 4 3 4 0 0
0 2 4 0 3 1 0 0 2 4 2 0 0 1 4 0 2 1 1 3 0
0 1 0 0 0 4 4 3 0 3 0 0 0 3 2 0 0 0 2 0 0
CROSSREFS
Row 4 of A186861.
Sequence in context: A275041 A109143 A260964 * A004383 A112390 A166757
KEYWORD
nonn,easy
AUTHOR
R. H. Hardin, Feb 27 2011
EXTENSIONS
a(34)-a(39) from J. Volkmar Schmidt, Sep 03 2023
STATUS
approved