This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A365553 #12 Oct 29 2023 20:36:01 %S A365553 5,6,8,20,861 %N 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. %C A365553 Definition is better understood when viewing the linked Desmos graph. %C A365553 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. %H A365553 Colin Linzer, <a href="https://www.desmos.com/calculator/4scocubtwl">Doodle Sequence: Intersections</a> (a Desmos graph). %F A365553 A recursive formula for an upper bound: %F A365553 a(n+1) <= (a(n)^4 - a(n)^2)/8 + (a(n) - a(n)^3)/4 which is equivalent to %F A365553 a(n+1) <= binomial(binomial(a(n),2),2) (proven). %F A365553 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. %Y A365553 Cf. A050534, A364725. %K A365553 nonn,hard,more %O A365553 1,1 %A A365553 _Colin Linzer_, Sep 08 2023