A383169 Triangle T(n,k) read by rows: For closed chains of j identical regular polygons with connecting inner vertices lying n vertices apart, the n-th row lists the possible j in descending order; n>=0, 1<=k<=d(8+4n).

10, 6, 4, 3, 14, 8, 6, 5, 4, 3, 18, 10, 6, 4, 3, 22, 12, 7, 6, 4, 3, 26, 14, 10, 8, 6, 5, 4, 3, 30, 16, 9, 6, 4, 3, 34, 18, 10, 6, 4, 3, 38, 20, 14, 11, 8, 6, 5, 4, 3, 42, 22, 12, 10, 7, 6, 4, 3, 46, 24, 13, 6, 4, 3, 50, 26, 18, 14, 10, 8, 6, 5, 4, 3

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 j > 2 there exists a chain with exactly j polygons.

A366872 gives the number of row elements.

The descending order in the definition was chosen with respect to the interconnection with A383168. For each n there are finitely many pairs (m,j) for j m-gons building closed chains. j are given by T(n,k) and the corresponding m are given by A383168(n,k).

m = 4 + 2n + (8+4n)/(j-2).

j = 2 + (8+4n)/(m-4-2n).

These two equations allow a computation of T(n,k) and A383168(n,k) from each other, see Formula.

Examples

			Triangle begins:
 10,  6,  4,  3;
 14,  8,  6,  5,  4, 3;
 18, 10,  6,  4,  3;
 22, 12,  7,  6,  4, 3;
 26, 14, 10,  8,  6, 5, 4, 3;
 30, 16,  9,  6,  4, 3;
 34, 18, 10,  6,  4, 3;
 38, 20, 14, 11,  8, 6, 5, 4, 3;
 42, 22, 12, 10,  7, 6, 4, 3;
 46, 24, 13,  6,  4, 3;
 50, 26, 18, 14, 10, 8, 6, 5, 4, 3;
 ...
The third row T(2,.) asserts that closed chains of identical regular polygons with connecting inner vertices lying 2 vertices apart can only be assembled with 18, 10, 6, 4 or 3 polygons.
		

Links

• Manfred Boergens, Closed chains of polygons.

Crossrefs

Cf. A366872, A383168.

Programs

Mathematica

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

Formula

T(n,k) = 2 + (k-th divisor of 8+4n in descending order).

T(n,k) = 2 + A027750(8+4n,A000005(8+4n)-k+1).

T(n,k) = 2 + (8+4n)/(A383168(n,k)-4-2n).

A383168(n,k) = 4 + 2n + (8+4n)/(T(n,k)-2).

T(n,1) = 10 + 4n.

T(n,2) = 6 + 2n.

T(n,2) = A383168(n,2).

T(n,3) = (2/3)*(7+2n) if n=1 mod 3, else = 4+n.

T(n,d(8+4n)) = 3 (last row elements).

T(n,d(8+4n)-1) = 4 (second to last row elements).

T(n,d(8+4n)-2) = 5 if n=1 mod 3, else = 6 (third last row elements).