A344311 Number of polygons formed outside a regular n-gon when every pair of vertices of the n-gon are joined by an infinite line.

0, 0, 0, 0, 5, 18, 49, 96, 198, 320, 550, 708, 1235, 1582, 2415, 2912, 4284, 4518, 7068, 8080, 11025, 12430, 16445, 16992, 23650, 26000, 32994, 35896, 44863, 42090, 59675, 64064, 77880, 83232, 99960, 102132, 126429, 133950, 157833, 166560, 194750, 191310, 237790, 249480, 287595, 301070

Offset: 1

Author

Scott R. Shannon, Jun 01 2021

Keywords

• nonn

Comments

See A344857 for other examples and images of the polygons.

Only finite polygons are counted. - N. J. A. Sloane, Jun 11 2021

Examples

			a(5) = 5 as the five connected vertices of a regular pentagon form five triangles outside the pentagon, along with sixteen polygons on the inside.
		

Crossrefs

Cf. A344857 (total number of polygons), A007678 (number of polygons inside the n-gon), A345025 (total number of polygons and open regions), A344938, A344857, A344866, A146213 (number of vertices).

Formula

For odd n, a(n) = (2*n^4 - 15*n^3 + 34*n^2 - 21*n)/24 (see A344857).

a(n) = A344857(n) - A007678(n).

Extensions

Edited by N. J. A. Sloane, Sep 12 2021