A386562 -id:A386562 - cp's OEIS frontend

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.

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.

A386560 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 finite regions created in the resulting graph.

Original entry on oeis.org

4, 72, 424, 1396, 3536, 7292, 14272, 24332, 39356, 59920, 91348, 128084, 182664, 245804, 323116, 418552, 547820, 684680, 869388, 1060892, 1289564, 1560920

Offset: 1

Scott R. Shannon, Jul 26 2025

Scott R. Shannon, Image for n = 1. In this and other images the integer coordinates 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

Cf. A386559 (vertices), A386561 (edges), A386562 (k-gons), A344993, A344857, A344279, A345459.

a(n) = A386561(n) - A386559(n) + 1 by Euler's formula.

A386561 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 finite edges created in the resulting graph.

Original entry on oeis.org

8, 136, 804, 2608, 6568, 13396, 26556, 45120, 73060, 110984, 171144, 238900, 344212, 462788, 607384, 786476, 1037772, 1293904, 1654432, 2013768, 2447312, 2965392

Offset: 1

Scott R. Shannon, Jul 26 2025

See A386559 and A386560 for images of the graphs.

Cf. A386559 (vertices), A386560 (regions), A386562 (k-gons), A347751, A344899, A344896, A345650.

a(n) = A386559(n) + A386560(n) - 1 by Euler's formula.

cp's OEIS Frontend

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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A386560 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 finite regions created in the resulting graph.

Views

Author

Keywords

Links

Crossrefs

Formula

A386561 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 finite edges created in the resulting graph.

Views

Author

Keywords

Comments

Crossrefs

Formula