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A165134 Number of directed Hamiltonian paths in the n X n knight graph.

%I A165134 #63 Feb 16 2025 08:33:11
%S A165134 1,0,0,0,1728,6637920,165575218320,19591828170979904
%N A165134 Number of directed Hamiltonian paths in the n X n knight graph.
%C A165134 Previous name was: Number of knight's paths visiting each square of an n X n chessboard exactly once.
%H A165134 Stefan Behnel, <a href="http://www.behnel.de/knight.html">The Knight's Paths</a>
%H A165134 A. Chernov, <a href="http://alex-black.ru/article.php?content=141">Open knight's tours</a>
%H A165134 Gheorghe Coserea, <a href="/A165134/a165134.txt">Solutions for 5x5 chessboard</a>
%H A165134 P. Hingston, G. Kendall, <a href="http://dx.doi.org/10.1109/CEC.2005.1554800">Enumerating knight's tours using an ant colony algorithm</a>, The 2005 IEEE Congress on Evolutionary Computation, 2 (2006), 1003-1010
%H A165134 G. Stertenbrink, <a href="http://magictour.free.fr/enum">Number of Knight's Tours</a>
%H A165134 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HamiltonianPath.html">Hamiltonian Path</a>
%H A165134 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KnightGraph.html">Knight Graph</a>
%e A165134 From _Gheorghe Coserea_, Oct 08 2016: (Start)
%e A165134 For n=5 the numbers in the table below give the number of knight's paths starting at the respective position on the 5 X 5 chessboard.  In total there are a(5) = 304*4 + 56*8 + 64 = 1728 solutions.
%e A165134     [1]  [2]  [3]  [4]  [5]
%e A165134 [1] 304    0   56    0  304
%e A165134 [2]   0   56    0   56    0
%e A165134 [3]  56    0   64    0   56
%e A165134 [4]   0   56    0   56    0
%e A165134 [5] 304    0   56    0  304
%e A165134 (End)
%Y A165134 Cf. Undirected Hamiltonian paths: A169696 (3 X n), A079137 (4 X n), A083386 (5 X n), A306281 (6 X n), A306283 (7 X n), A308131 (n X n).
%Y A165134 Cf. A001230, A118067, A306282.
%K A165134 nonn,hard,more
%O A165134 1,5
%A A165134 [No name given] (c.candide(AT)free.fr), Sep 04 2009
%E A165134 a(7) from Guenter Stertenbrink, added by _Alex Chernov_, Sep 01 2013
%E A165134 a(1)=1, a(2)=0 prepended by _Max Alekseyev_, Sep 22 2013
%E A165134 a(8) from _Alex Chernov_, May 10 2014
%E A165134 Name made more precise by _Eric W. Weisstein_, Apr 14 2019