This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A383169 #10 May 01 2025 19:59:08
%S A383169 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,
%T A383169 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,
%U A383169 3,46,24,13,6,4,3,50,26,18,14,10,8,6,5,4,3
%N 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).
%C A383169 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.
%C A383169 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.)
%C A383169 For every j > 2 there exists a chain with exactly j polygons.
%C A383169 A366872 gives the number of row elements.
%C A383169 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).
%C A383169 m = 4 + 2n + (8+4n)/(j-2).
%C A383169 j = 2 + (8+4n)/(m-4-2n).
%C A383169 These two equations allow a computation of T(n,k) and A383168(n,k) from each other, see Formula.
%H A383169 Manfred Boergens, <a href="https://github.com/maboerg/Closed-chains-of-polygons">Closed chains of polygons</a>.
%F A383169 T(n,k) = 2 + (k-th divisor of 8+4n in descending order).
%F A383169 T(n,k) = 2 + A027750(8+4n,A000005(8+4n)-k+1).
%F A383169 T(n,k) = 2 + (8+4n)/(A383168(n,k)-4-2n).
%F A383169 A383168(n,k) = 4 + 2n + (8+4n)/(T(n,k)-2).
%F A383169 T(n,1) = 10 + 4n.
%F A383169 T(n,2) = 6 + 2n.
%F A383169 T(n,2) = A383168(n,2).
%F A383169 T(n,3) = (2/3)*(7+2n) if n=1 mod 3, else = 4+n.
%F A383169 T(n,d(8+4n)) = 3 (last row elements).
%F A383169 T(n,d(8+4n)-1) = 4 (second to last row elements).
%F A383169 T(n,d(8+4n)-2) = 5 if n=1 mod 3, else = 6 (third last row elements).
%e A383169 Triangle begins:
%e A383169 10, 6, 4, 3;
%e A383169 14, 8, 6, 5, 4, 3;
%e A383169 18, 10, 6, 4, 3;
%e A383169 22, 12, 7, 6, 4, 3;
%e A383169 26, 14, 10, 8, 6, 5, 4, 3;
%e A383169 30, 16, 9, 6, 4, 3;
%e A383169 34, 18, 10, 6, 4, 3;
%e A383169 38, 20, 14, 11, 8, 6, 5, 4, 3;
%e A383169 42, 22, 12, 10, 7, 6, 4, 3;
%e A383169 46, 24, 13, 6, 4, 3;
%e A383169 50, 26, 18, 14, 10, 8, 6, 5, 4, 3;
%e A383169 ...
%e A383169 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.
%t A383169 Table[2 + Sort[Divisors[8 + 4 n], Greater], {n, 0, 10}]//Flatten
%Y A383169 Cf. A366872, A383168.
%K A383169 nonn,tabf
%O A383169 1,1
%A A383169 _Manfred Boergens_, Apr 18 2025