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<106, ∑ z(n) = 7894453.
Find ∑ z(n) for 0<n<1017.
齐肯多夫表示
斐波那契数列的每一项都由前两项相加而得。
从1和2开始，前10项是：1、2、3、5、8、13、21、34、55、89。
每一个正整数可以唯一地写成斐波那契数列中非连续项的和。例如，100 = 3 + 8 + 89。
这样的和被称为数的齐肯多夫表示。
对于任意整数n>0，记z(n)为n的齐肯多夫表示中的项数。
因此，z(5) = 1，z(14) = 2，z(100) = 3，等等。
此外，对于0<n<106，∑ z(n) = 7894453。
对于0<n<1017，求∑ z(n)。