Three Geometric Sequences

[Ali Sada]

Hi Everyone,

Hope all is well. I would really appreciate it if you could tell me if any of the following three sequences is suitable of the OEIS.

1. Maximum number of triangles from a self-intersecting unit chain.
a1(n) is the maximum number of triangles formed by a self-intersecting planar chain of n connected unit segments. The chain may turn in any direction, and the construction is cumulative rather than restarted for each n.

2. Greedy sequence based on successive Pythagorean triangles.
This sequence is defined by constructing right triangles while giving priority, whenever possible, to the smallest unused positive integers.
We begin with
1,2,4,5.

Since 1+2=3, these lengths allow the construction of the 3,4,5 triangle.

Next, we use 3+5=8 and we draw a line of length 6 perpendicular to 8, and then we draw a segment of 10 forming the triangle 6,8,10.

So, the sequence continues
1,2,4,5,3,6,10,…

(Although 8 is available for the construction, it is not entered into the sequence, because it arises only as a sum of two earlier numbers.)

More generally, the rules are:
a) At each step, give priority to the smallest unused positive integer.
b) Previously listed terms may be extended to produce auxiliary lengths for the construction.
c) Auxiliary lengths obtained only by addition are not themselves entered into the sequence.
d) The segments can be extended but overlaps are not allowed.

The main question is whether every positive integer eventually appears in this way.

3. Maximum number of squares from an alternating chain
a3(n) is the maximum number of squares formed by a planar chain whose segment lengths alternate
1,2,1,2,1,2,…

The chain may self-intersect, and the goal is to maximize the number of distinct squares formed by the first n segments.

Best,

Ali