A344993 Number of polygons formed when every pair of vertices of a row of n adjacent congruent rectangles are joined by an infinite line.

0, 4, 20, 68, 168, 368, 676, 1184, 1912, 2944, 4292, 6152, 8456, 11484, 15164, 19624, 24944, 31508, 39076, 48212, 58656, 70672, 84284, 100192, 117888, 138100, 160580, 185796, 213568, 245008, 279116, 317424, 359280, 405124, 454868, 509264, 567640, 631988, 701228, 776032, 855968, 943260, 1035844

Offset: 0

Author

Scott R. Shannon and N. J. A. Sloane, Jun 05 2021

Keywords

• nonn

Comments

The number of polygons formed inside the rectangles is A306302(n), while the number of polygons formed outside the rectangles is 2*A332612(n+1).

The number of open regions, those outside the polygons with unbounded area and two edges that go to infinity, for n >= 1 is given by 2*n^2 + 4*n + 6 = A255843(n+1).

Like A306302(n) is appears only 3-gons and 4-gons are generated by the infinite lines.

Examples

			a(1) = 4 as connecting the four vertices of a single rectangle forms four triangles inside the rectangle. Twelve open regions outside these triangles are also formed.
a(2) = 20 as connecting the six vertices of two adjacent rectangles forms two quadrilaterals and fourteen triangles inside the rectangles while also forming four triangles outside the rectangles, giving twenty polygons in total. Twenty-two open regions outside these polygons are also formed.
See the linked images for further examples.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10000
• Scott R. Shannon, Image for n = 1. In this and other images the vertices at the corners of all rectangles are highlighted as white dots while the outer open regions, which are not counted, are darkened.
• 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.
• Scott R. Shannon, Image for n = 7.

Crossrefs

See A347750 and A347751 for the numbers of vertices and edges in the finite part of the corresponding graph.

Cf. A332612 (half the number of polygons outside the rectangles), A306302 (number of polygons inside the rectangles), A255843.

Programs

Python

from sympy import totient
def A344993(n): return 2*n*(n+1) + 2*sum(totient(i)*(n+1-i)*(2*n+2-i) for i in range(2,n+1)) # Chai Wah Wu, Aug 21 2021

Formula

a(n) = 2*A332612(n+1) + A306302(n) = 2*Sum_{i=2..n, j=1..i-1, gcd(i,j)=1} (n+1-i)*(n+1-j) + Sum_{i=1..n, j=1..n, gcd(i,j)=1} (n+1-i)*(n+1-j) + n^2 + 2*n.

a(n) = 2*n*(n+1) + 2*Sum_{i=2..n} (n+1-i)*(2*n+2-i)*phi(i). - Chai Wah Wu, Aug 21 2021