A344857 Number of polygons formed when every pair of vertices of a regular n-gon are joined by an infinite line.

0, 0, 1, 4, 16, 42, 99, 176, 352, 540, 925, 1152, 2016, 2534, 3871, 4608, 6784, 6984, 11097, 12580, 17200, 19250, 25531, 26016, 36576, 39988, 50869, 55076, 68992, 63570, 91575, 97920, 119296, 127024, 152881, 155088, 193104, 203946, 240787, 253360, 296800, 289044, 362061, 378884, 437536, 456918

Offset

1,4

Comments

For odd n, a(n) is given by the equation in the Formula section below. See also A344866. For even n no such equation is currently known, although one similar to the general formula for the number of polygons inside an n-gon, see A007678, likely exists.

The number of open regions, those outside the polygons with unbounded area and two edges that go to infinity, for n >= 3 is given by n*(n-1) = A002378(n-1).

See A345025 for the total number of all areas, both polygons and open regions.

Links

Scott R. Shannon, Table of n, a(n) for n = 1..100

J. F. Rigby, Multiple intersections of diagonals of regular polygons, and related topics, Geom. Dedicata 9 (1980), 207-238.

Scott R. Shannon, Image for n = 3. In this and other images the n-gon vertices are highlighted as white dots while the outer open regions, which are not counted, are darkened. The key for the edge-number coloring is shown at the top-left of the image.

Scott R. Shannon, Image for n = 4.

Scott R. Shannon, Image for n = 5.

Scott R. Shannon, Image for n = 6.

Scott R. Shannon, Image for n = 7.

Scott R. Shannon, Image for n = 8.

Scott R. Shannon, Image for n = 9.

Scott R. Shannon, Image for n = 10.

Scott R. Shannon, Image for n = 11.

Scott R. Shannon, Image for n = 12.

Scott R. Shannon, Image for n = 13.

Scott R. Shannon, Image for n = 14.

Scott R. Shannon, Image for n = 15.

Scott R. Shannon, Image for n = 19.

Scott R. Shannon, Image for n = 21.

Scott R. Shannon, Image for n = 24.

Alexander Sidorenko, Explicit Formulas for Odd-Indexed Terms in A344899, A146212, and A344857.

Formula

For odd n, a(n) = (n^4 - 7*n^3 + 19*n^2 - 21*n + 8)/8 = (n-1)^2*(n^2-5*n+8)/8. This was conjectured by Scott R. Shannon and proved by Alexander Sidorenko on Sep 10 2021 (see link). - N. J. A. Sloane, Sep 12 2021

See also A344866.

a(n) = A344311(n) + A007678(n).

Example

a(1) = a(2) = 0 as no polygon can be formed by one or two connected points.
a(3) = 1 as the connected vertices form a triangle, while the six regions outside the triangle are open.
a(4) = 4 as the four connected vertices form four triangles inside the square. Twelve open regions outside these polygons are also formed.
a(5) = 16 as the five connected vertices form eleven polygons inside the regular pentagon while also forming five triangles outside the pentagon, giving sixteen polygons in total. Twenty open regions outside these polygons are also formed.
a(6) = 42 as the six connected vertices form twenty-four polygons inside the regular hexagon while also forming eighteen polygons outside the hexagon, giving forty-two polygons in total. Thirty open regions outside these polygons are also formed.
See the linked images above for further examples.

Crossrefs

Cf. A344311 (number outside the n-gon), A007678 (number inside the n-gon), A345025 (total number of regions), A344866 (number for odd n), A146212 (number of vertices), A344899 (number of edges), A344938 (number of k-gons), A002378 (number of open regions for (n-1)-gon).

Bisections: A344866, A347320.

Sequence in context: A323847 A114211 A188124 * A190090 A227012 A034131

Adjacent sequences: A344854 A344855 A344856 * A344858 A344859 A344860

Keyword

nonn

Author

Scott R. Shannon, May 30 2021

Status

approved