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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..22.

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.

Formula

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

Crossrefs

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

Sequence in context: A292228 A195579 A296369 * A061184 A118004 A281232

Adjacent sequences: A386556 A386557 A386558 * A386560 A386561 A386562

Keyword

nonn,more

Author

Scott R. Shannon, Jul 26 2025

Status

approved