Not your standard base

A number in the Fibonacci base is defined as follows:

• The number is composed of a string of 1's and 0's (like binary)

• No two consecutive 1's occur in any number (e.g. $101000101_{F}$ is a number in the Fibonacci base.)

• The place value of each digit from left to right is corresponding to the Fibonacci sequence

• Conversion to base 10 requires multiplying the place value by it's corresponding digit. e.g.

$\begin{aligned} 100101_{F} &= 1 \times F_1 + 0 \times F_2 + 1 \times F_3 + 0 \times F_4 + 0 \times F_5 + 1 \times F_6\\ &= 1 + 3 + 13\\&=16 \end{aligned}$

Do all natural numbers have a representation in the Fibonacci base?

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The Fibonacci sequence is defined as:

$F_1 = 1, F_2 = 2, F_n = F_{n-1} + F_{n-2} \quad \text{for } n > 2$