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

1,3

Comments

See A344857 for other examples and images of the polygons.

Links

Table of n, a(n) for n=1..44.

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.

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.).

Example

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.

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.

Sequence in context: A245361 A247340 A067354 * A148877 A148878 A148879

Adjacent sequences: A344896 A344897 A344898 * A344900 A344901 A344902

Keyword

nonn

Author

Scott R. Shannon, Jun 02 2021

Status

approved