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A366872
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Number of closed chains of identical regular polygons with connecting inner vertices lying n vertices apart.
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0
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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, 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, 8, 6, 20, 6, 8, 18, 9, 12, 16, 6, 10, 12, 16, 6, 18, 6, 8, 18
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OFFSET
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0,1
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COMMENTS
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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.
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.)
Geometrical reasoning shows that for each n there are finitely many (not zero) chains with the described properties.
a(n) is the number of these chains and equals d(8+4n), the number of divisors of 8+4n.
For every m > 4 there exists a chain of m-gons.
For every j > 2 there exists a chain with exactly j polygons.
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LINKS
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FORMULA
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a(n) > 5, with the exceptions a(0) = 4 and a(2) = 5.
a(n) = 6 iff n = 6 or n + 2 is an odd prime.
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EXAMPLE
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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.
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.
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MATHEMATICA
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Table[{n, Length[Divisors[8+4 n]]}, {n, 0, 107}] // TableForm
(With additional output describing the chains:)
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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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