1,4

For n points on a circle, there are floor(n/2) distinct line segment lengths. Hence an upper bound for a(n) is the number of compositions of n-1 into floor(n/2) nonnegative parts, which is A099578(n-2). Conjecture: the upper bound is attained if n is prime. There are A052558(n-2) paths to be considered. - T. D. Noe, Jan 09 2007 [Edited by Petros Hadjicostas, Jul 19 2018]

Samuel C. Gutekunst, Circulant TSP: Vertices of the Edge-Length Polytope and Superpolynomial Lower Bounds, arXiv:2506.10758 [cs.DM], 2025. See pp. 2-3, 17-18.

Sean A. Irvine, Java program (github)

Brendan D. McKay and Tim Peters, Paths through equally spaced points on a circle, arXiv:2205.06004 [math.CO], 2022.

For n=4 the 3 lengths are: 3 boundary edges (length 3), edge-diagonal-edge (2 + sqrt(2)) and diagonal-edge-diagonal (1 + 2*sqrt(2)).

For n=5, the 4 edges of the path may include 0,...,4 diagonals, so a(5)=5.

Cf. A007874 (similar, but with n line segments), A052558, A099578.

See A352568 for the multisets of line lengths.

Sequence in context: A032619 A193066 A193070 * A352568 A058580 A161682

Adjacent sequences: A030074 A030075 A030076 * A030078 A030079 A030080

nonn,nice

Daniel Lurie Gittelson, Dec 11 1999

a(13) - a(16) from T. D. Noe, Jan 09 2007

Removed unnecessary mention of dihedral group from definition. - N. J. A. Sloane, Apr 02 2022

The terms a(1) to a(15) have been verified by Sean A. Irvine and a(1) to a(16) by Brendan McKay. - N. J. A. Sloane, Apr 02 2022

a(17) to a(37) from Brendan McKay, May 14 2022

approved