KöMaL 2024 A893

In a text editor program, initially there is a footprint symbol (L) that we want to multiply. Unfortunately, our computer has been the victim of a hacker attack, and only two functions are working: Copy and Paste, each costing $1$ Dürer dollar to use. Using the Copy function, we can select one or more consecutive symbols from the existing ones, and the computer memorizes their number. When using the Paste function, the computer adds as many new footprint symbols to the sequence as were selected in the last Copy. If no Copy has been done yet, Paste cannot be used. Let $D(n)$ denote the minimum number of Dürer dollars required to obtain exactly $n$ footprint symbols. Prove that for any positive integer $k$, there exists a positive integer $N$ such that $D(N)={D(N+1)}+1={D(N+2)}={D(N+3)}+1={D(N+4)}=\ldots ={D(N+2k-1)}+1={D(N+2k)}$.