A108053 Maximum number of diagonals of a regular n-gon that meet at a non-center point.

0, 0, 2, 2, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2

Offset

3,3

Comments

Starting at a(13) = 2, sequence is periodic with period 30.

Links

Paolo Xausa, Table of n, a(n) for n = 3..10000

Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, arXiv:math/9508209 [math.MG], 1995-2006.

Sequences formed by drawing all diagonals in regular polygon

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Formula

From Paolo Xausa, May 11 2023: (Start)

a(n) = 0 if n <= 4.

For n > 4:

a(n) = 2 if n is odd or n = 6;

a(n) = 3 if n != 6 is even but not divisible by 6;

a(n) = 4 if n = 12;

a(n) = 5 if n != 12 is divisible by 6 but not 30;

a(n) = 7 if n is divisible by 30. (End)

Example

In a 30-gon, there are non-center points where 7 diagonals meet, but no more than 7. Hence a(30) = 7.

Mathematica

LinearRecurrence[PadLeft[{1}, 30], {0, 0, 2, 2, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5}, 120] (* Ray Chandler, Aug 27 2015 - adapted to new data by Paolo Xausa, May 15 2023 *)
PadRight[{0, 0, 2, 2, 2, 3, 2, 3, 2, 4}, 120, {2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3}] (* Harvey P. Dale, Jun 20 2021 - adapted to new data by Paolo Xausa, May 15 2023 *)

Crossrefs

Cf. A006561, A007678.

Sequence in context: A292585 A175872 A143117 * A076221 A029235 A289052

Adjacent sequences: A108050 A108051 A108052 * A108054 A108055 A108056

Keyword

easy,nonn

Author

David W. Wilson, Jun 01 2005

Extensions

a(4), a(6) and a(12) corrected by Paolo Xausa, May 11 2023

Status

approved