Algorithmic Information Theory defines the extent to which a

sequence of numbers is random by the length of the shortest

algorithm (i.e. programme) that outputs it. So "11111111111111" could be output by a little programme like

*
*

for i=1 to 15

{print "1"}

while the shortest

programme to output "346357538323523627567" might actually be

*
*

print "346357538323523627567"

so the second sequence is "more random" than the first. In fact it defines a "random sequence" as one for which the shortest algorithm

**is** just "

*print* the sequence".

You can make this idea more concrete by considering a

Turing machine made to output the sequence rather than just a

random programming language.

Note that this ties in nicely with our common understanding of randomness - if there's a nice pattern there, then you can exploit that to compress the sequence into a little algorithm, and hence it is not by this definition random.