Property for repetitive functions

There's a $k$ with $f(k)=a_1,\ldots,f(k+n-1)=a_n$, $f(k+n)=1$.

There's a $k'$ with $f(k')=a_1,\ldots,f(k'+n-1)=a_n$, $f(k'+n)=2$ etc.

For every finite sequence there are infinitely many finite distinct sequences that extend it.