Each new term in the Fibonacci sequence is generated by adding the previous two terms.
Starting with $1$ and $2$, the first $10$ terms will be: $1, 2, 3, 5, 8, 13, 21, 34, 55, 89$.
Every positive integer can be uniquely written as a sum of nonconsecutive terms of the Fibonacci sequence. For example, $100 = 3 + 8 + 89$.
Such a sum is called the Zeckendorf representation of the number.
For any integer $n \gt 0$, let $z(n)$ be the number of terms in the Zeckendorf representation of $n$.
Thus, $z(5) = 1$, $z(14) = 2$, $z(100) = 3$ etc.
Also, for $0 \lt n \lt 10^6$, $\sum z(n) = 7894453$.
Find $\sum z(n)$ for $0 \lt n \lt 10^{17}$.
Solution
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