1,2

A general purpose matrix-transfer method can be used to compute values up to a(4). Using a diagonal sweep from one corner to the opposite corner will help to reduce the number of states. - Andrew Howroyd, Dec 20 2015

Schram & Schiessel (see Links) quote a different result for a(4): 27747833510015886 undirected Hamiltonian walks, which would double to 55495667020031772 directed Hamiltonian walks. However, that number is not divisible by 8 and thus cannot be correct. - Arun Giridhar, Dec 15 2015

Raoul D. Schram and Helmut Schiessel, Exact enumeration of Hamiltonian walks on the 4x4x4 cube and applications to protein folding, Journal of Physics A: Mathematical and Theoretical, vol 46 (2013), 485001.

Raoul D. Schram and Helmut Schiessel, Corrigendum: Exact enumeration of Hamiltonian walks on the 4x4x4 cube and applications to protein folding, Journal of Physics A: Mathematical and Theoretical, vol 49 (2016), 369501.

Jamie Shepard, Solvability and Difficulty of the Snake Puzzle in the Cube and its Topological Variants, Honors Thesis, Andrews Univ. (2024) Art. No. 290.

Eric Weisstein's World of Mathematics, Grid Graph

Eric Weisstein's World of Mathematics, Hamiltonian Path

For n = 1, there is a trivial Hamiltonian path of length 0.

For n = 2, the 144 paths fall in three different equivalence classes. Two of the three classes can be derived by taking a Hamiltonian cycle on a cube and deleting a single edge. The third class is a spiral path that ends at the opposite corner from its starting point.

Cf. A003763, A096969.

Sequence in context: A318197 A362214 A393875 * A003837 A013863 A371624

Adjacent sequences: A193343 A193344 A193345 * A193347 A193348 A193349

nonn,hard,more,bref

Eric W. Weisstein, Jul 23 2011

a(4) from Andrew Howroyd, Nov 15 2015

a(1) corrected by Arun Giridhar, Dec 20 2015

approved