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A328184
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Denominator of time taken for a vertex of a rolling regular n-sided polygon to reach the ground.
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1
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4, 8, 20, 12, 28, 16, 12, 20, 44, 24, 52, 28, 20, 32, 68, 36, 76, 40, 28, 44, 92, 48, 100, 52, 36, 56, 116, 60, 124, 64, 44, 68, 140, 72, 148, 76, 52, 80, 164, 84, 172, 88, 60, 92, 188, 96, 196, 100, 68, 104, 212, 108, 220, 112, 76, 116, 236, 120, 244, 124, 84
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OFFSET
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3,1
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COMMENTS
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Given an n-sided regular polygon "rolling" on a flat surface with constant angular velocity, a(n) is the denominator of the ratio: [("time" taken for any one vertex to move from highest point to lowest point) / ("time" taken for polygon to finish one complete turn)] := b(n).
Lim_{n->infinity} b(n) = 1/2 (can be easily proved).
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LINKS
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FORMULA
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a(n) = denominator((n - 1) / (2*n)) for even n; a(n) = denominator((2*n - 3) / (4*n)) for odd n.
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EXAMPLE
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For n = 3, a(3) = denominator of ((2*3-3)/4*n) = denominator of (3/12) = denominator of (1/4) = 4.
a(4) = 8 since it takes 3/8 of a full revolution of a square for a vertex to go from the highest point to the lowest point. When the vertex is at its highest position the square will be oriented at 45 degrees to the plane.
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MATHEMATICA
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Array[Denominator[(2 (# - 1) - Mod[#, 2])/(4 #)] &, 61, 3] (* Michael De Vlieger, Oct 06 2019 *)
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PROG
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(PARI) a(n) = {denominator((2*(n-1) - n%2)/(4*n))} \\ Andrew Howroyd, Oct 06 2019
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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