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Problem 297


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)。