A365553 Starting with a plane on which two parallel lines and two additional lines have been drawn such that the four lines form two noncongruent isosceles triangles, a(n) is the total number of intersections on the plane after the n-th step, where each step consists of drawing lines that connect every intersection of two lines. If more than 2 lines intersect at the same point it is only counted once.

5, 6, 8, 20, 861

Offset: 1

Author

Colin Linzer, Sep 08 2023

Keywords

• nonn
• hard
• more

Comments

Definition is better understood when viewing the linked Desmos graph.

The parity of a term is the parity of the number of intersections along the median from the base of the isosceles triangles. This is because the median is an axis of symmetry of the plane, the number of intersections not on it must be even.

Links

• Colin Linzer, Doodle Sequence: Intersections (a Desmos graph).

Crossrefs

Cf. A050534, A364725.

Formula

A recursive formula for an upper bound:

a(n+1) <= (a(n)^4 - a(n)^2)/8 + (a(n) - a(n)^3)/4 which is equivalent to

a(n+1) <= binomial(binomial(a(n),2),2) (proven).

The proof of the above formula comes from the fact that if there are o points on a graph, then at most (o^2-o)/2 lines that can be drawn between them. If there are m lines on a graph, then there are at most (m^2-m)/2 intersections between them; substituting and simplifying leads to the former upper limit.