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 A366872 #19 May 06 2025 10:28:55
%S A366872 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,
%T A366872 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,
%U A366872 8,6,20,6,8,18,9,12,16,6,10,12,16,6,18,6,8,18
%N A366872 Number of closed chains of identical regular polygons with connecting inner vertices lying n vertices apart.
%C A366872 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.
%C A366872 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 A366872 Geometrical reasoning shows that for each n there are finitely many (not zero) chains with the described properties.
%C A366872 a(n) is the number of these chains and equals d(8+4n), the number of divisors of 8+4n.
%C A366872 For every m > 4 there exists a chain of m-gons. The possible m for each n are given by A383168.
%C A366872 For every j > 2 there exists a chain with exactly j polygons. The possible j for each n are given by A383169.
%H A366872 Manfred Boergens, <a href="https://github.com/maboerg/Closed-chains-of-polygons">Closed chains of polygons</a>.
%F A366872 a(n) = A000005(8+4n).
%F A366872 a(n) > 5, with the exceptions a(0) = 4 and a(2) = 5.
%F A366872 a(n) = 6 iff n = 6 or n + 2 is an odd prime.
%e A366872 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.
%e A366872 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.
%t A366872 Table[{n, Length[Divisors[8+4 n]]}, {n, 0, 107}] // TableForm
%t A366872 (With additional output describing the chains:)
%t A366872 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}]
%Y A366872 Cf. A000005, A383168, A383169.
%K A366872 nonn
%O A366872 0,1
%A A366872 _Manfred Boergens_, Oct 26 2023