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A344993 Number of polygons formed when every pair of vertices of a row of n adjacent congruent rectangles are joined by an infinite line. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

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.

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

EXAMPLE

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.

PROG

(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

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.

Sequence in context: A302317 A319779 A287244 * A123613 A006740 A291526

Adjacent sequences:  A344990 A344991 A344992 * A344994 A344995 A344996

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified May 16 22:18 EDT 2022. Contains 353724 sequences. (Running on oeis4.)