A385525 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 of the same length.

327080, 84, 52, 32, 18, 24, 24, 24, 24, 24, 18, 24, 24, 24, 24, 24, 24, 24, 24, 30, 24, 30, 30, 24, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 36, 30, 36, 36, 30, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 40, 36, 36, 40, 36, 40, 42

Offset: 2

Author

Charles L. Hohn, Jul 30 2025

Keywords

• nonn

Comments

Other than a(2) whose walk is comprised of 8 diagonal segments, all known terms are produced by 3- or 4-segment walks, including some with examples of both. It is conjectured that this holds true for all n >= 3.

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.

Removing the constraint that the diagonal segments must all have the same length gives A386251. All such walks in 2 dimensions, by diagonal segment length rather than total walk length, gives A386671.

Examples

			a(2) = 327080 because [3636, 40723] + [8844, 39917] + [11603, -39204] + [38076, -14893] + [-37523, -16236] + [35844, -19667] + [-34387, -22116] + [-26093, 31476] = [0, 0] and 8 segments * length 40885 = 327080, which is the smallest example for n = 2.
a(3) = 84: [16, 11, 8] + [-13, 4, 16] + [-8, -19, -4] + [5, 4, -20] = [0, 0, 0] and 4 * 21 = 84.
a(4) = 52: [8, 8, 5, 4] + [-9, -6, 6, 4] + [-7, -4, -10, 2] + [8, 2, -1, -10] = [0, 0, 0, 0] and 4 * 13 = 52.
		

Links

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

Crossrefs

Cf. A386251, A386671.

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