Is there a divergent sequence such that for every n in N it is possible to find n consecutive twos somewhere in the sequence.

Question

Is there a divergent sequence such that for every n in N it is possible to find n consecutive twos somewhere in the sequence.

My thoughts are sequences as : { 1 , 2, 2, 2, 2, 3, 2, 2, 2, 4, ...} or something like the whole |N and then we follow with inf. twos .

I dont know how to write this as a function/sequence form like an = .... .

Help appreciated! :)

Answer 1

Take an infinite sequence of $2$'s and intersperse it with the terms of some divergent sequence, each time increasing the distance.

Like

$$\begin{cases}T_{n(n+1)/2}&=S_n,\\T_{k,k\ne n(n+1)/2}&=2.\end{cases}$$