ISI2016-PCB-A-3

Question

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 $a_i's).$

Answer 1

I think a(n)=a(n-1)+2; a0=1

Answer 2

 

Figure 4:

Source :

Discrete mathematics and its applications by Kenneth H. Rosen