This is a small puzzle I've been playing with for the past couple of days:
For some length N, how many unique sequences of digits can be created when any 'rotation' of the digits is considered as the same sequence?
Examples of equivalent sequences under this definition (that is, each column of values is considered one sequence):
N=3
123 445 777
312 544
231
N=4
1234
4123
3412
2341
Note: In the above example for N=4, 1324 is considered distinct from 1234, because 1324 cannot be reached by rotating 1234.
"Unique rotating sequences" of the kind you describe are called necklaces. According to the linked article on Wikipedia, the number of $k$-ary necklaces of length $N$ is given by $$ \frac{1}{N}\sum_{d\;\vert\;N}\varphi(d)k^{N/d}, $$ where $k=10$ if you're considering strings of the digits $0...9$, the sum is over the divisors of $N$, and $\varphi(d)$ is Euler's totient function, or the number of positive integers less than $d$ that are relatively prime to $d$.