A366872 Number of closed chains of identical regular polygons with connecting inner vertices lying n vertices apart.

4, 6, 5, 6, 8, 6, 6, 9, 8, 6, 10, 6, 8, 12, 7, 6, 12, 6, 10, 12, 8, 6, 12, 9, 8, 12, 10, 6, 16, 6, 8, 12, 8, 12, 15, 6, 8, 12, 12, 6, 16, 6, 10, 18, 8, 6, 14, 9, 12, 12, 10, 6, 16, 12, 12, 12, 8, 6, 20, 6, 8, 18, 9, 12, 16, 6, 10, 12, 16, 6, 18, 6, 8, 18

Offset: 0

Author

Manfred Boergens, Oct 26 2023

Keywords

• nonn

Comments

Consider j identical regular polygons, assembled into a circular closed chain. Two neighboring polygons share an edge and two vertices, the "inner" one lying in the interior of the chain. The interior is a j-pointed star with equal edges.

n is introduced in order to partition the set of chains into finite subsets. Two neighboring star points are separated by n vertices; there the star has reflex angles. (With n=0, regular polygons are considered as stars with no reflex angles.)

Geometrical reasoning shows that for each n there are finitely many (not zero) chains with the described properties.

a(n) is the number of these chains and equals d(8+4n), the number of divisors of 8+4n.

For every m > 4 there exists a chain of m-gons. The possible m for each n are given by A383168.

For every j > 2 there exists a chain with exactly j polygons. The possible j for each n are given by A383169.

Examples

			a(0) = 4 is the number of chains of identical regular polygons which have an interior regular polygon, namely 10 pentagons, 6 hexagons, 4 octagons, 3 dodecagons.
a(1) = 6 is the number of chains of identical regular polygons which have an interior proper star with identical edges, namely 14 heptagons, 8 octagons, 6 nonagons, 5 decagons, 4 dodecagons, 3 18-gons.
		

Links

• Manfred Boergens, Closed chains of polygons.

Crossrefs

Cf. A000005, A383168, A383169.

Programs

Mathematica

Table[{n, Length[Divisors[8+4 n]]}, {n, 0, 107}] // TableForm
(With additional output describing the chains:)
Do[Print["n = ", n, " a(n) = ", Length[Divisors[8+4 n]]]; d = Divisors[8+4 n]; le = Length[d]; Do[t1 = d[[i]]; t2 = (8+4 n)/d[[i]]; Print["m = ", t1+4+2 n, " j = ", t2+2], {i,le}], {n, 0, 19}]

Formula

a(n) = A000005(8+4n).

a(n) > 5, with the exceptions a(0) = 4 and a(2) = 5.

a(n) = 6 iff n = 6 or n + 2 is an odd prime.