Counting climbs of n steps in strides of one or two

Someone walks up a staircase of $n$ steps, and each stride covers either one step or two steps. How many different stride sequences take them from the bottom to exactly the top?

$F_{n+1}$, the $(n+1)$-th Fibonacci number (with $F_1=F_2=1$). The last stride starts from step $n-1$ or step $n-2$, so the count $a_n$ satisfies $a_n=a_{n-1}+a_{n-2}$ with $a_1=1$, $a_2=2$, giving $1,2,3,5,8,\dots$