A378831 The minimum number of diagonal lines required to cover all vertices created when the n outer vertices of a regular n-gon are connected by diagonal lines.

1, 2, 2, 4, 5, 10, 11, 21

Offset: 2

Author

Scott R. Shannon, Dec 08 2024

Keywords

• nonn
• more

Comments

See A007569 for the total number of vertices created by the diagonal lines of a regular n-gon.

The terms a(9)..a(13) are less than or equal to 21, 25, 38, 41, 59 respectively.

Note that these numbers are not always the same as the minimum number of required lines if the lines can be arbitrarily placed. The first known occurrence where the numbers differ is for a(7) = 10 where one can use eight diagonal lines plus one off-diagonal line to cover two vertices in such a way that all vertices can be covered with only 9 lines.

Better upper bounds for a(10)..a(13) are 23, 36, 38, 56, with a(10), a(11) being almost certainly the correct values. - Scott R. Shannon, Dec 10 2024

Links

• Scott R. Shannon, Image showing one example of 11 diagonals covering all 57 vertices of the octagon. The covering diagonals are shown as thicker white lines.

Crossrefs

Cf. A007569, A373813, A373811.

Extensions

a(9) added by Scott R. Shannon, Dec 10 2024