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).

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

Author

Manfred Boergens, Apr 18 2025

Keywords

• nonn
• tabf

Comments

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.

Examples

			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.
		

Links

• Manfred Boergens, Closed chains of polygons.

Crossrefs

Cf. A047244, A366872, A383169.

Programs

Mathematica

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

Formula

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).