Can any sequence of numbers ultimately be determined by an algorithm?

Can any sequence of numbers ultimately be determined by an algorithm?

I am talking about absolutely any sequence. As an example; If one were to pluck a thousand random numbers from their head, is it be possible that a specific algorithm (although maybe extremely long and complicated to work out) could apply to predict this sequence and its ordering? If it is true, would this mean the concept of random does not exist?
Thanks for the responses. I am thinking that for each set of numbers, an infinite number of possible algorithms exists, in order to accommodate the prediction of the next number in the sequence, which can range from 0 to ±infinity.

I am left thinking, though, that a possible algorithm may contain an infinite number of logical steps. This would therefore render it as being illegal, as, by definition, it must have a finite number of steps.

Whatever the case, it seems as though random, in the sense that a set of numbers chosen can have no predictable pattern between them, does not truly exist. Anyone able to assist or correct further?