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

Author

Scott R. Shannon, Jun 06 2021

Keywords

• nonn

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.

Examples

			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.
		

Links

• 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.

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.

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).