Recursive (Counting) Intermediate

Problem - 2480

Joe wants to write $1$ to $n$ in a $1 \times n$ grid. The number 1 can be written in any grid, while the number $2$ must be written next to $1$ (can be at either side) so that these two numbers are together. The number 3 must be written next to this two-number block. This process goes on. Every new number written must stay next to the existing number block. How many different ways can Joe fill this $1 \times n$ grid?


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