## Zeckendorf Representation

### Problem 297

Published on Friday, 18th June 2010, 05:00 pm; Solved by 1788; Difficulty rating: 35%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}.