A386251 Consider the graph whose vertices are the points of the n-dimensional cubic lattice with points connected by all integer-length diagonals that traverse all n dimensions and do not intersect intermediate points. a(n) is the total length of the shortest possible closed walk in this graph via noncongruent diagonals.

60, 28, 20, 22, 18, 16, 18, 18, 16, 18, 18, 18, 20, 20, 18, 20, 20, 20, 22, 22, 22, 24, 24, 22, 24, 24, 24, 26, 24, 24, 26, 26, 26, 26, 26, 26, 28, 28, 28, 28, 30, 28, 30, 30, 30, 30, 30, 30, 32, 32, 32, 32, 32, 32, 32, 34, 34, 34, 34, 34, 36, 34, 34, 36, 36

Offset: 2

Author

Charles L. Hohn, Jul 16 2025

Keywords

• nonn
• nice

Comments

It is provable that all such walks must be even in total length. It is also provable that 3-segment closed walks are impossible for n < 6, and conjectured that a(n) for all n >= 6 are produced by 3-segment walks.

For n = 2, the walk segments are the hypotenuses of noncongruent primitive Pythagorean triangles.

The offset is 2, because even though the graph could be defined in dimension 1 (the vertices would be the points of Z, with each point connected to its two neighbors), it would not contain any closed paths.

Adding a constraint that the diagonal segments must all have the same length gives A385525.

Examples

			a(2) = 60 because [3, 4] + [5, 12] + [-15, 8] + [7, -24] = [0, 0] and segment lengths 5 + 13 + 17 + 25 = 60, which is the smallest example for n = 2.
a(3) = 28: [2, 2, 1] + [-3, -2, -6] + [-7, 4, 4] + [8, -4, 1] = [0, 0, 0] and 3 + 7 + 9 + 9 = 28.
a(4) = 20: [1, 1, 1, 1] + [-1, -2, -2, -4] + [-5, -3, -1, 1] + [5, 4, 2, 2] = [0, 0, 0, 0] and 2 + 5 + 6 + 7 = 20.
		

Links

• Charles L. Hohn, a(2), graphical view
• Charles L. Hohn, a(3), graphical view, animated

Crossrefs

Cf. A385525.

Cf. A020882 (diagonals in 2 dimensions), A096910 (diagonals in 3 dimensions).