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 A383168 #13 May 09 2025 01:31:31
%S A383168 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,
%T A383168 18,20,24,36,15,16,18,21,28,42,17,18,20,24,32,48,19,20,21,22,24,27,30,
%U A383168 36,54,21,22,24,25,28,30,40,60,23,24,26,33,44,66
%N 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).
%C A383168 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 A383168 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 A383168 For every m > 4 there exists a chain of m-gons.
%C A383168 A366872 gives the number of row elements.
%C A383168 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).
%C A383168 j = 2 + (8+4n)/(m-4-2n).
%C A383168 m = 4 + 2n + (8+4n)/(j-2).
%C A383168 These two equations allow a computation of T(n,k) and A383169(n,k) from each other, see Formula.
%H A383168 Manfred Boergens, <a href="https://github.com/maboerg/Closed-chains-of-polygons">Closed chains of polygons</a>.
%F A383168 T(n,k) = 4+2n + (k-th divisor of 8+4n in ascending order).
%F A383168 T(n,k) = 4+2n + A027750(8+4n,k).
%F A383168 T(n,k) = 4+2n + (8+4n)/(A383169(n,k)-2).
%F A383168 A383169(n,k) = 2 + (8+4n)/(T(n,k)-4-2n).
%F A383168 T(n,1) = 5+2n.
%F A383168 T(n,2) = 6+2n.
%F A383168 T(n,2) = A383169(n,2).
%F A383168 T(n,3) = 7+2n if n=1 mod 3, else = 8+2n.
%F A383168 T(n,3) = A047244(5+n).
%F A383168 T(n,d(8+4n)) = 12+6n (last row elements).
%F A383168 T(n,d(8+4n)-1) = 8+4n (second to last row elements).
%F A383168 T(n,d(8+4n)-2) = (10/3)*(2+n) if n=1 mod 3, else = 3*(2+n) (third last row elements).
%e A383168 Triangle begins:
%e A383168 5, 6, 8, 12;
%e A383168 7, 8, 9, 10, 12, 18;
%e A383168 9, 10, 12, 16, 24;
%e A383168 11, 12, 14, 15, 20, 30;
%e A383168 13, 14, 15, 16, 18, 20, 24, 36;
%e A383168 15, 16, 18, 21, 28, 42;
%e A383168 17, 18, 20, 24, 32, 48;
%e A383168 19, 20, 21, 22, 24, 27, 30, 36, 54;
%e A383168 21, 22, 24, 25, 28, 30, 40, 60;
%e A383168 23, 24, 26, 33, 44, 66;
%e A383168 25, 26, 27, 28, 30, 32, 36, 40, 48, 72;
%e A383168 ...
%e A383168 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.
%t A383168 Table[4 + 2*n + Divisors[8 + 4 n], {n, 0, 10}]//Flatten
%Y A383168 Cf. A047244, A366872, A383169.
%K A383168 nonn,tabf
%O A383168 1,1
%A A383168 _Manfred Boergens_, Apr 18 2025