A345025 Number of regions formed when every pair of vertices of a regular n-gon are joined by an infinite line.

1, 2, 7, 16, 36, 72, 141, 232, 424, 630, 1035, 1284, 2172, 2716, 4081, 4848, 7056, 7290, 11439, 12960, 17620, 19712, 26037, 26568, 37176, 40638, 51571, 55832, 69804, 64440, 92505, 98912, 120352, 128146, 154071, 156348, 194436, 205352, 242269, 254920, 298440, 290766, 363867, 380776, 439516

Offset

1,2

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.

See A344857 for further examples and images of the regions.

Links

Table of n, a(n) for n=1..45.

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 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 = 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.

Formula

Formula for odd n: a(n) = (n^4 - 7*n^3 + 27*n^2 - 29*n + 8)/8 (see A344857).

For n >= 3, a(n) = A344857(n) + A002378(n-1).

Example

a(2) = 2 as an infinite line connecting two points cuts space into two unbounded regions.
a(3) = 7 as the three connected points of the 3-gon form one closed triangle along with six outer unbounded areas, seven regions in total.
a(4) = 16 as the four connected points of the 4-gon form four closed triangle inside the square along with twelve outer unbounded areas, sixteen regions in total.

Crossrefs

Cf. A344857 (number of polygons), A344311 (number polygons outside the n-gon), A007678 (number polygons inside the n-gon), A002378 (number of open regions for (n-1)-gon), A146212 (number of vertices), A344866, A344938.

Sequence in context: A259966 A283500 A097442 * A329324 A131405 A269963

Adjacent sequences: A345022 A345023 A345024 * A345026 A345027 A345028

Keyword

nonn

Author

Scott R. Shannon, Jun 06 2021

Status

approved