A347750 -id:A347750 - cp's OEIS frontend

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

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

nonn

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.

			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.

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.

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.

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

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

A386559 Place a point on the integer coordinates, up to |n|, along all four axial directions on a Cartesian plane, and then join an infinite straight line between every pair of points: a(n) is the number of points where lines intersect in the resulting graph.

Original entry on oeis.org

5, 65, 381, 1213, 3033, 6105, 12285, 20789, 33705, 51065, 79797, 110817, 161549, 216985, 284269, 367925, 489953, 609225, 785045, 952877, 1157749, 1404473

Offset: 1

Scott R. Shannon, Jul 26 2025

It appears that for n >= 3 the intersection that is furthest from the origin is formed by the crossing of the lines y = n/(n-1)*x + n and y = (n-1)/(n-2)*x - (n-1), along with the seven other symmetrically equivalent intersections. These intersections have a distance from the origin of approximately sqrt(8)*n^3 as n -> infinity.

Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.

Cf. A386560 (regions), A386561 (edges), A386562 (k-gons), A146212, A347750, A344657, A345649.

a(n) = A386561(n) - A386560(n) + 1 by Euler's formula.

A347751 Number of finite edges in the graph formed when every pair of vertices of a row of n adjacent congruent rectangles are joined by an infinite line.

Original entry on oeis.org

0, 8, 36, 124, 300, 664, 1200, 2108, 3388, 5232, 7568, 10852, 14892, 20288, 26704, 34540, 43812, 55400, 68584, 84684, 103004, 124216, 147888, 175820, 206788, 242424, 281560, 325708, 374148, 429416, 489000, 556412, 629804, 710536, 797280, 892564, 994588, 1107744, 1228432, 1359292, 1498788

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Sep 12 2021

nonn

See A344993 and A347750 for images of the rectangles.

			a(1) = 8 as connecting the four vertices of a single rectangle forms four new edges inside the rectangle, giving a total of 4 + 4 = 8 total edges.
a(2) = 36 as connecting the six vertices of two adjacent rectangles forms twenty-two edges inside the rectangles while also forming eight edges outside the rectangles. The total number of edges is then 6 + 22 + 8 = 36.

Cf. A344993 (number of polygons), A347750 (number of intersections), A331757 (number of edges on or inside the rectangles).

a(n) = A344993(n) + A347750(n) - 1.

A092275 Draw a line through every pair of points with coordinates (x, 1) and (x', 2) with x, x' in 1..n, and then count the number of intersection points above the line y = 2.

Original entry on oeis.org

0, 2, 11, 29, 69, 125, 224, 361, 563, 809, 1164, 1597, 2188, 2871, 3714, 4701, 5956, 7359, 9099, 11067, 13365, 15879, 18884, 22201, 26056, 30219, 34951, 40117, 46073, 52441, 59716, 67599, 76320, 85569, 95790, 106701, 118902

Offset: 2

N. J. A. Sloane, Feb 17 2004

nonn

Joseph O'Rourke, Art Gallery Theorems and Algorithms, Oxford Univ. Press, 1987. See section 8.6, p. 223.

Cf. A331755, A347750.

a(n) = (A347750(n-1) - A331755(n)) / 2. - Andrey Zabolotskiy, Jan 17 2024

New name and terms a(10) and beyond (using the formula) added by Andrey Zabolotskiy, Jan 18 2024

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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.

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A386559 Place a point on the integer coordinates, up to |n|, along all four axial directions on a Cartesian plane, and then join an infinite straight line between every pair of points: a(n) is the number of points where lines intersect in the resulting graph.

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A347751 Number of finite edges in the graph formed when every pair of vertices of a row of n adjacent congruent rectangles are joined by an infinite line.

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A092275 Draw a line through every pair of points with coordinates (x, 1) and (x', 2) with x, x' in 1..n, and then count the number of intersection points above the line y = 2.

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