A383169 -id:A383169 - cp's OEIS frontend

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

Manfred Boergens, Oct 26 2023

nonn

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.

			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.

Manfred Boergens, Closed chains of polygons.

Cf. A000005, A383168, A383169.

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}]

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.

A383168 Triangle T(n,k) read by rows: For closed chains of identical regular m-gons with connecting inner vertices lying n vertices apart, the n-th row lists the possible m in ascending order; n>=0, 1<=k<=d(8+4n).

Original entry on oeis.org

5, 6, 8, 12, 7, 8, 9, 10, 12, 18, 9, 10, 12, 16, 24, 11, 12, 14, 15, 20, 30, 13, 14, 15, 16, 18, 20, 24, 36, 15, 16, 18, 21, 28, 42, 17, 18, 20, 24, 32, 48, 19, 20, 21, 22, 24, 27, 30, 36, 54, 21, 22, 24, 25, 28, 30, 40, 60, 23, 24, 26, 33, 44, 66

Offset: 1

Manfred Boergens, Apr 18 2025

Consider j identical regular m-gons, 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.)
For every m > 4 there exists a chain of m-gons.
A366872 gives the number of row elements.
This sequence is interconnected with A383169. For each n there are finitely many pairs (m,j) for j m-gons building closed chains. m are given by T(n,k) and the corresponding j are given by A383169(n,k).
j = 2 + (8+4n)/(m-4-2n).
m = 4 + 2n + (8+4n)/(j-2).
These two equations allow a computation of T(n,k) and A383169(n,k) from each other, see Formula.

			Triangle begins:
  5,  6,  8, 12;
  7,  8,  9, 10, 12, 18;
  9, 10, 12, 16, 24;
 11, 12, 14, 15, 20, 30;
 13, 14, 15, 16, 18, 20, 24, 36;
 15, 16, 18, 21, 28, 42;
 17, 18, 20, 24, 32, 48;
 19, 20, 21, 22, 24, 27, 30, 36, 54;
 21, 22, 24, 25, 28, 30, 40, 60;
 23, 24, 26, 33, 44, 66;
 25, 26, 27, 28, 30, 32, 36, 40, 48, 72;
 ...
The third row T(2,.) asserts that regular 9-gons, 10-gons, 12-gons, 16-gons and 24-gons are the only regular polygons which can be assembled to a closed chain with connecting inner vertices lying 2 vertices apart.

Manfred Boergens, Closed chains of polygons.

Cf. A047244, A366872, A383169.

Table[4 + 2*n + Divisors[8 + 4 n], {n, 0, 10}]//Flatten

T(n,k) = 4+2n + (k-th divisor of 8+4n in ascending order).
T(n,k) = 4+2n + A027750(8+4n,k).
T(n,k) = 4+2n + (8+4n)/(A383169(n,k)-2).
A383169(n,k) = 2 + (8+4n)/(T(n,k)-4-2n).
T(n,1) = 5+2n.
T(n,2) = 6+2n.
T(n,2) = A383169(n,2).
T(n,3) = 7+2n if n=1 mod 3, else = 8+2n.
T(n,3) = A047244(5+n).
T(n,d(8+4n)) = 12+6n (last row elements).
T(n,d(8+4n)-1) = 8+4n (second to last row elements).
T(n,d(8+4n)-2) = (10/3)*(2+n) if n=1 mod 3, else = 3*(2+n) (third last row elements).

cp's OEIS Frontend

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

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