cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

Scott R. Shannon, May 30 2021

Keywords

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.

Examples

			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.

Links

Crossrefs

Cf. A344311 (number of finite regions 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.

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).

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.

Original entry on oeis.org

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

Views

Author

Scott R. Shannon, Jun 01 2021

Keywords

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

A343755 Number of regions formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.

Original entry on oeis.org

7, 30, 144, 474, 1324, 2934, 5797, 10614, 17424, 27480, 41845, 61602, 85711, 120120, 159213, 207798, 269668, 349272, 434878, 545496, 661764, 804582, 973471, 1174980, 1374646, 1631304, 1908768, 2218254, 2560198, 2976486, 3378985, 3887796, 4405671, 4995240, 5617689, 6322878
Offset: 1

Views

Author

Scott R. Shannon, Jun 28 2021

Keywords

Comments

The count of regions includes both the closed bounded polygons and the open unbounded areas surrounding these polygons with two edges that go to infinity. The number of unbounded areas appears to equal 6*(n^2 - n + 1).
See A344279 for further examples and images of the regions.

Examples

			a(1) = 7 as the three connected vertices of a triangle form one polygon along with six outer unbounded areas, seven regions in total.
a(2) = 30 as when the three vertices and three edges points are connected they form twelve polygons, all inside the triangle, along with eighteen outer unbounded areas, thirty regions in total.
a(2) = 144 as when the three vertices and six edges points are connected they form one hundred two polygons, seventy-five inside the triangle and twenty-seven outside, along with forty-two outer unbounded areas, one hundred forty-four regions in total.

Links

  • Scott R. Shannon, Image for n = 1. In this and other images the triangle's vertices are highlighted as white dots while the outer open regions are cross-hatched. The key for the edge-number coloring is shown at the top-left of the image. Note the edge count for open areas also includes the two infinite edges
  • Scott R. Shannon, Image for n = 2.
  • Scott R. Shannon, Image for n = 3.
  • Scott R. Shannon, Image for n = 4.
  • Scott R. Shannon, Image for n = 5.
  • Scott R. Shannon, Image for n = 6.

Crossrefs

Cf. A344279 (number of polygons), A344657 (number of vertices), A344896 (number of edges), A346446 (number of k-gons), A092867 (number polygons inside the triangle), A121205, A345025.

Formula

Conjectured formula: a(n) = A344279(n) + 6*(n^2 - n + 1).
Conjectured formula: a(n) = A344279(n) + A121205(n-1), for n>=7.

A383461 Number of vertices in graph G_n formed by taking a regular n-gon with all its chords extended to infinity (the n-th graph in A344857) and inverting it in its circumscribing circle.

Original entry on oeis.org

4, 5, 16, 37, 92, 145, 334, 471, 892, 901, 1964, 2185, 3796, 3969, 6682, 5563, 10964, 11141, 17032, 17293, 25324, 21913, 36326, 36479, 50572, 50485, 68644, 51661, 91172, 90753, 118834, 118355, 152356, 139861, 192512, 191445, 240124, 238481
Offset: 3

Views

Author

Scott R. Shannon and N. J. A. Sloane, Jun 02 2025

Keywords

Comments

Inverting a point or a line in a circle C with center O and radius r is a classical operation in geometry (Coxeter, Section 6.3; Pedoe, pp. 4-9). Every point A inside C except O itself has an inverse point A' outside the circle; A' lies on the line OA and satisfies |OA|*|OA'| = r^2. The inverse of the center O is undefined.
If a line L passes through O its inverse is L itself. If L is not a diameter of C, and meets C in two points A and B, the inverse of L is the circle through O, A, and B.
Theorem: G_n has A345025(n) regions. If n is even then n of these regions are infinite, otherwise there is a single infinite region.
The initial versions of the illustrations were made by NJAS using GeoGebra. The colored versions were added later by SRS using a Java program. These have greater resolution and include information about the vertex and region counts.

References

  • H. S. M. Coxeter, Introduction to Geometry, Wiley, 1961.
  • D. Pedoe, Circles: A Mathematical View, Dover, 1979.

Links

Crossrefs

Cf. A146212, A344857, A345025, A383462.

Formula

a(n) = A146212(n) + (n mod 2).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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