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Consider following inputs:
1,1,2,3,5,8 – it isn’t random
2,4,8,16,32 – this neither
4,1,2,11,5,9– this one looks like random-sequence
I would like to ask if is there such algorithm to prove if input is random or it isn’t?
No, there is no such prove – if you have perfectly random numbers, the probability of each sequence of length n is equal.
However, there are statistical tests to asses the quality of a random number generator, which is probably what you are looking for. See Diehard tests.