Number of 10 digit numbers having digits 0,1,2 so that no two 0s are consecutive.
The recurrence for $b_n$ is $b_n = 2 c_{n-1} = 2b_{n-1} + 2a_{n-1} = 2b_{n-1}+2b_{n-2}$ since given any number of length $n-1$ you can construct two numbers of length $n$ ending with $1$ or $2$.