User blog:GamesFan2000/GamesFan's Graph Function

So, there are n dots on a piece of paper. The rules for this function are as follows:

The dots MUST be a consistent distance apart. Meaning that the distance between two dots that are adjacent to each other have the same distance between each other as two other adjacent dots. The dots can only be at angles in multiples of five from other adjacent dots.

The dots must be placed in such a way that they are part of a regular polygon or symmetrical shape with at most 2n vertices.

You can use at most 2n lines to build your graph. A line must have a size that is a multiple of the consistent distance of the dots, or it must connect two dots at least. A line can go through dots and not be broken. A line can also go through another line and not be broken. Multiple lines can go through the same dot. A dot MUST be at an end of a line or the line MUST have a dot at halfway between the ends of a line, or the line MUST be connected at a vertice to at least one other line. The lines can't extend beyond the outermost dots unless they create a shape, or they can be 'unfolded' into a symmetrical shape. The lines must be angled in such a way that the angle is a multiple of 5 from their ends. A line can be a regular circle, which counts as just one line, but a dot must be on it and the imaginary line from the dot and the opposite side must have an angle from its vertices that is a multiple of 5. Multiple circles can be connected to the same dot, and circles of different sizes will not be considered validly different unless they go through more dots or dots in different positions, or unless another circle was already connected at the EXACT same position as the larger circle, in which case the larger circle MUST have a circumference twice as big as the smaller circle. If three or more circles are connected at the same position, they must grow exponentially, with the second circle being twice as big as the first, the third being twice as big as the second, and so on. Each graph must be sufficiently different in a way other than size, like dots being in different configurations, or lines being at different angles, etc.

Let F(n) be the number of different graphs that can be built with at most n dots. This function will specifically called GFG(n).