Luca Alexander - cp's OEIS frontend

User: Luca Alexander

Luca Alexander has authored 2 sequences.

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

Luca Alexander, Oct 06 2019

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).

			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.

Cf. A328185 (numerators).

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

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

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

A328185 Numerators associated with A328184.

Original entry on oeis.org

1, 3, 7, 5, 11, 7, 5, 9, 19, 11, 23, 13, 9, 15, 31, 17, 35, 19, 13, 21, 43, 23, 47, 25, 17, 27, 55, 29, 59, 31, 21, 33, 67, 35, 71, 37, 25, 39, 79, 41, 83, 43, 29, 45, 91, 47, 95, 49, 33, 51, 103, 53, 107, 55, 37, 57, 115, 59, 119, 61, 41, 63, 127, 65, 131, 67

Offset: 3

Luca Alexander, Oct 06 2019

Geometric Interpretation: Given n-sided regular polygon "rolling" on a flat surface with constant angular velocity, a(n) is the numerator 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 proven).

			For n = 3, a(3) = numerator of ((2*3-3)/4*n) = numerator of (3/12) = numerator of (1/4) = 1.

Luca Alexander, about 100000 terms

Cf. A328184 (denominators).

Array[Numerator[(2 (# - 1) - Mod[#, 2])/(4 #)] &, 66, 3] (* Michael De Vlieger, Oct 06 2019 *)

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

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

cp's OEIS Frontend

User: Luca Alexander

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

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