What is the relation between this binary number with no two 1 side by side and fibonacci sequence? I saw this pattern of binary numbers with constraints first number should be 1 , and two 1's cannot be side by side.
Now as an example
1 = 1
10 = 1
100,101 = 2
1000,1001,1010 = 3
10000,10001, 10010, 10100, 10101 = 5 

Strangely I see the numbers we can form of this binary numbers with $n $digit is the$ n$th Fibonacci number , at least for for the first 5 Fibonacci number. How Can we show that it is true for nth number ? and how is this happening?

It's well known (and easy to prove) that the Fibonacci numbers give the number of ways you can tile a $1\times n$ strip with $1\times1$ squares and $1\times2$ dominoes -- see, for example, the introductory section of this paper by Benjamin, Quinn, and Su.  Ignoring, the initial $1$ in your binary numbers, everything that follows can be interpreted as a string of squares, each with a $0$ written on them, and dominoes, each with a $01$ (in that order), written on them.
