# Fi Binary Number [closed]

A Fi-binary number is a number that contains only 0 and 1. It does not contain any leading 0. And also it does not contain 2 consecutive 1. The first few such number are 1, 10, 100, 101, 1000, 1001, 1010, 10000, 10001, 10010, 10100, 10101 and so on. I have n. How to calculate the nth Fi-binary number ?

## closed as off-topic by user230715, Claude Leibovici, kjetil b halvorsen, drhab, DidAug 20 '15 at 9:59

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• Actually, I don't have to calculate the $n$th FiBinary number, you do. So, what steps have you taken to do so? – Gerry Myerson Aug 20 '15 at 7:15
• My question is , How to calculate nth Fi-binary number ? n can be 10^9, so what sill be the 10^9th Fi-binary number ? – M S Hossain Aug 20 '15 at 7:19
• I know what your question is. My question remains: what steps have you taken toward answering your question? – Gerry Myerson Aug 20 '15 at 7:21
• I attempt a naive step , check the number one by one, it is Fi-binary or not? if it is , take the number in the queue of Fi-binary number. if queue size is n then the last value of queue is answer. but it should be optimized . It has a pattern to calculate , I think . I am asking for this pattern. – M S Hossain Aug 20 '15 at 7:28
• The question is interesting. Describe your efforts within the question, and show your attempts towards reaching a solution on your own. Thus, you will increase your chances of being taken more seriously, and perhaps even getting some proper answers to your question. – barak manos Aug 20 '15 at 7:51

## 1 Answer

Hint: Let $F_k$ denote the $k$-th Fibonacci number; i.e., $F_0=0$, $F_1=1$, and $F_k=F_{k-1}+F_{k-2}$ for all integers $k\geq 2$. Every positive integer $n$ can uniquely be written as a sum of nonconsecutive pairwise distinct Fibonacci numbers $F_k$'s with $k\geq 2$. If $n=\sum_{j=1}^m\,F_{k_j}$ is such a representation, then the $n$-th Fi-binary number is $\sum_{j=1}^m\,10^{k_j-2}$. If you are talking about the base-$2$ representation, then replace $10$ by $2$.

• but how the idea about the final result come ? explain please @Batominovski – M S Hossain Aug 20 '15 at 8:32