Counting the number of binary strings of length m with no consecutive 1s (RR). How to improve it?


I am new to Mathematica and I am trying to solve this problem of counting the number of binary strings of a certain length m, as far as no consecutive 1s are there.

For instance m = 3, my recurrence relation should give 5 i.e. 000, 001, 101, 100, 010.

I started like this with initial seeds: n > 0; a[ 1] = 1, a[2] = 3, a[3] = 5 and then in RSolve I did:

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Is there a better way to improve it so that I can use the output in a plot? Currently, I cannot as my solution says it cannot be used as function.

Thanks.