A392000 Array read by antidiagonals: A(k, n) is the number of undirected closed Hamiltonian paths on the k X n knight graph.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9862, 0, 0, 0, 0, 0, 0, 0, 16, 0, 44202, 1067638, 1067638, 44202, 0, 16, 0, 0

Offset

1,50

Comments

A Hamiltonian knight path is closed if the start and the end point of the path are adjacent points on the knight graph, i.e., if the final square visited is one valid knight's move away from the starting square.

For the general case (allowing open knight's tours) see A390833.

References

Donald E. Knuth, Hamiltonian paths and cycles, Prefascicle 8a of The Art of Computer Programming (work in progress, 2025).

Links

Table of n, a(n) for n=1..78.

George Jelliss, Introducing Knight's Tours, 2004.

Donald E. Knuth, Adventures with Knight's Tours, Christmas lecture Dec. 2025.

Peter Luschny, Hamiltonian Paths On Knight Graphs.

Eric Weisstein's World of Mathematics, Hamiltonian Path.

Eric Weisstein's World of Mathematics, Knight Graph.

Wikipedia, Knight's tour.

Index entries for sequences related to graphs, Hamiltonian

Example

Array starts:
  [k\n] 1 2  3  4      5         6            7               8                 9
  --------------------------------------------------------------------------------
  [1]  0, 0, 0, 0,     0,        0,           0,              0,                0, ... A000004
  [2]  0, 0, 0, 0,     0,        0,           0,              0,                0, ... A000004
  [3]  0, 0, 0, 0,     0,        0,           0,              0,                0, ... A169764
  [4]  0, 0, 0, 0,     0,        0,           0,              0,                0, ... A000004
  [5]  0, 0, 0, 0,     0,        8,           0,          44202,                0, ... A175855
  [6]  0, 0, 0, 0,     8,     9862,     1067638,       55488142,       3374967940, ... A175881
  [7]  0, 0, 0, 0,     0,  1067638,           0,    34524432316,                0, ... A193054
  [8]  0, 0, 0, 0, 44202, 55488142, 34524432316, 13267364410532, 7112881119092574, ... A193055
  [9]  0, 0, 0, 0,     0,3374967940,          0,7112881119092574,               0, ... A391999
.
A closed knight's tour on a 5 X 6 board: a1 -> c2 -> e1 -> d3 -> e5 -> c6 -> a5 -> c4 -> b6 -> a4 -> b2 -> d1 -> e3 -> d5 -> b4 -> a6 -> c5 -> e6 -> d4 -> e2 -> c1 -> a2 -> c3 -> e4 -> d6 -> b5 -> a3 -> b1 -> d2 -> b3.

Crossrefs

Cf. A000004, A169764, A175855, A175881, A193054, A193055, A391999, A001230, A390833 (all paths).

Sequence in context: A098432 A392625 A390833 * A197623 A291692 A127583

Adjacent sequences: A391997 A391998 A391999 * A392001 A392002 A392003

Keyword

nonn,tabl

Author

Peter Luschny, Dec 26 2025

Status

approved