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