Problem 297

Zeckendorf Representation

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`>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`n`10^{6}, ∑ `z`(`n`) = 7894453.

Find ∑ `z`(`n`) for 0`n`10^{17}.

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