A328184 Denominator of time taken for a vertex of a rolling regular n-sided polygon to reach the ground.

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

Offset

3,1

Comments

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).

Links

Table of n, a(n) for n=3..63.

Formula

a(n) = denominator((n - 1) / (2*n)) for even n; a(n) = denominator((2*n - 3) / (4*n)) for odd n.

Example

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.

Mathematica

Array[Denominator[(2 (# - 1) - Mod[#, 2])/(4 #)] &, 61, 3] (* Michael De Vlieger, Oct 06 2019 *)

Prog

(PARI) a(n) = {denominator((2*(n-1) - n%2)/(4*n))} \\ Andrew Howroyd, Oct 06 2019

Crossrefs

Cf. A328185 (numerators).

Sequence in context: A215112 A340948 A265108 * A332367 A273143 A273174

Adjacent sequences: A328181 A328182 A328183 * A328185 A328186 A328187

Keyword

nonn,frac

Author

Luca Alexander, Oct 06 2019

Status

approved