In one of her videos (at 0:46) Vihart muses about this problem. Given a wiggly plastic snake with $k$ links, how many valid and unique shapes can be created out of the snake.
A shape is valid if it does not contain a loop. For similarity, both rotational and reflections symmetry is allowed.(consider head and tail like any other link)
For example,
_ _ _| | and | |_ are same _ |_ _| | and _| are also same
and so on.
So, for a snake with two links there are only two possible shapes
_ _ and _|
for a three-link snake
_ _ _ _ _ , _ _ | , _ | , _|
I don't know a closed form mathematical expression for $f(k)$, the number of uniques shapes that can be created with a snake of $k$ links. But I did write a program to find $f(k)$. The numbers I got for different values of $k$ are given below. I was wondering if anyone can comment on the correctness of the numbers or know of a closed form expression for $f(k)$.
links shapes 0 0 1 1 2 2 3 4 4 9 5 22 6 56 7 147 8 388 9 1047 10 2806 11 7600 12 20437 13 55313 14 148752 15 401629 16 1078746