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

Author

David W. Wilson, Jun 01 2005

Keywords

• easy
• nonn

Comments

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

Examples

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

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).

Crossrefs

Cf. A006561, A007678.

Programs

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 *)

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)

Extensions

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