A393552 Number of distinct line sets in an n-gon connecting vertices and midpoints, disallowing same-edge connections, up to rotations and reflections.

20, 8712, 107439744, 23456309586688, 84327973045825475584, 4951760157143491424441862144, 4726143985013034288556068553395716096, 73075081866545145910200683026960909940431650816, 18260659593852162222067751059472598012305414686778089799680

Offset

3,1

Comments

There are n vertices and n midpoints. The total number of lines connecting vertices and midpoints is 2*n*(n-2).

Links

Paolo Xausa, Table of n, a(n) for n = 3..41

Rhys Feltman, Python script that calculates number of configurations of the lines connecting vertices and midpoints in an n-gon

Rhys Feltman, List of the 20 configurations of lines connecting vertices and midpoints for a triangle

Formula

Where c is the number of cycles in each of the 2n symmetries, a(n) = Sum_{2^c}/(2n).

a(n) = (1/2) * ( 2^floor((n*(2*n-3)-3)/2) + 2^floor((n*(2*n-3)-2)/2) + (1/n) * Sum_{d|n} phi(d)*2^((2*n - 3 - d mod 2)*n/d) ). - Andrew Howroyd, Mar 02 2026

Example

For n = 3, a(3) = 20, since there are 20 distinct ways you can connect vertices and midpoints without using the same edge, including the empty and full configurations.

Mathematica

A393552[n_] := (2^Quotient[# - 3, 2] + 2^Quotient[# - 2, 2] + DivisorSum[n, EulerPhi[#]*2^((2*n - 3 - Mod[#, 2])*n/#) &]/n)/2 & [n*(2*n - 3)];
Array[A393552, 10, 3] (* Paolo Xausa, Mar 09 2026 *)

Prog

(PARI)
r(n) = {my(m=2*n*(n-2)); if(n%2, 2^((m+n-1)/2), 3*2^((m+n)/2-2))}
c(n) = {my(m=2*n*(n-2)); sumdiv(n, d, eulerphi(d)*2^if(n%2, m/d, (m-n)/d + gcd(d, 2)*n/d))/n}
a(n) = {(c(n) + r(n))/2} \\ Andrew Howroyd, Feb 21 2026

Crossrefs

Cf. A191563.

Sequence in context: A013725 A307914 A293286 * A250021 A203308 A109122

Adjacent sequences: A393549 A393550 A393551 * A393553 A393554 A393555

Keyword

nonn

Author

Rhys Feltman, Feb 20 2026

Status

approved