Greeetings,
Does anyone know any number sequences such that the sum of the first n of these is not equal to a sum (with repition allowed )of any n of these first n numbers .Thanks.

I'm not following the question. Suppose the first 5 numbers add to a particular sum, call it S5. Then if you set n=5 the sum of the first n numbers is obviously S5. Example: if the sequence is 1,2,3,4,5,6,7,8,. and if n = 5 then Sn = 1+2+3+4+5 = 15. This of course equals the sum of 5 of the first 5 numbers. So how could any sequence of numbers satisfy the condition you're asking about?