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A bit string is called legitimate if it contains no consecutive zeros $, e.g., 0101110$ is legitimate, where as $10100111$ is not. Let $a_n$ denote the number of legitimate bit strings of length $n$. Define $a_0=1$. Derive a recurrence relation for $a_n ( i.e.,$  express $a_n$ in terms of the preceding $a_i's).$
in Combinatory by Veteran (105k points)
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I think a(n)=a(n-1)+2; a0=1
by Junior (511 points)
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Figure 4:

Source :

Discrete mathematics and its applications by Kenneth H. Rosen

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