I have an infine sequence where at the end of which the ones are written. Then till infinity we shall do the next procedure: for each segment with ends a and b (inside which the numbers are absent) we shall write in the middle the number a + b
Sequence buiding: {${1, 1}$} $-$ {${1, 2, 1}$} $-$ {${1, 3, 2, 3, 1}$} $-$ ${1, 4, 3, 5, 2, 5, 3, 4, 1}$
How many times the positive integer n will be written on a segment?
One observation: the number of occurrences of $n$ in the limit is determined after $n$ iterations (where the first iteration is the original sequence $\{1,1\}$. You can see this inductively. It's clearly true for $n=1$; now suppose that at some point at or after the $n^{\mathrm{th}}$ iteration we get another $n$ in the sequence. Then there must have been two adjacent values adding to $n$; since those are both less than $n$, they were added by the $(n-1)^{\mathrm{st}}$ iteration. So we are at the $n^{\mathrm{th}}$ iteration.