A344899 Number of polygon edges formed when every pair of vertices of a regular n-gon are joined by an infinite line.

0, 1, 3, 8, 30, 78, 189, 320, 684, 1010, 1815, 2052, 3978, 4718, 7665, 8576, 13464, 12546, 22059, 23720, 34230, 36542, 50853, 47928, 72900, 76466, 101439, 105560, 137634, 115230, 182745, 188672, 238128, 245378, 305235, 294948, 385614, 395390, 480909, 491840, 592860, 544950, 723303, 737528

Offset: 1

Author

Scott R. Shannon, Jun 02 2021

Keywords

• nonn

Comments

See A344857 for other examples and images of the polygons.

Examples

			a(3) = 3 as the connected vertices form a triangle with three edges. Six infinite edges between the outer regions are also formed but these are not counted.
a(5) = 30 as the five connected vertices form a pentagon with fives lines along the pentagon's edges, fifteen lines inside forming eleven polygons, and ten lines outside forming another five triangles. In all these sixteen polygons form thirty edges. Twenty infinite edges between the outer regions are also formed.
		

Links

• J. F. Rigby, Multiple intersections of diagonals of regular polygons, and related topics, Geom. Dedicata 9 (1980), 207-238.
• Alexander Sidorenko, Explicit Formulas for Odd-Indexed Terms in A344899, A146212, and A344857.

Crossrefs

Cf. A344907 (number of edges for odd n), A344857 (number of polygons), A146212 (number of vertices), A344866, A344311, A007678, A331450, A344938.

Bisections: A344907, A347322.

Formula

Conjectured formula odd n: a(n) = (n^4 - 7*n^3 + 17*n^2 - 11*n)/4 = (n-1)*n*(n^2-6*n+11)/4.

This formula is correct: see the Sidorenko link. - N. J. A. Sloane, Sep 12 2021

See also A344907.

a(n) = A344857(n) + A146212(n) - 1 (Euler's theorem.).