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Problem 340

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Crazy Function

For fixed integers a, b, c, define the crazy function F(n) as follows:
F(n) = n - c for all n > b
F(n) = F(a + F(a + F(a + F(a + n)))) for all n ≤ b

Also, define S(a, b, c) = $\sum_{n=0}^{b}F(n)$.

For example, if a = 50, b = 2000 and c = 40, then F(0) = 3240 and F(2000) = 2040.
Also, S(50, 2000, 40) = 5204240.

Find the last 9 digits of S(21⁷, 7²¹, 12⁷).

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疯狂函数

对于给定的整数a、b、c，定义疯狂函数F(n)如下：
对于所有n > b，F(n) = n - c
对于所有n ≤ b，F(n) = F(a + F(a + F(a + F(a + n))))

此外，定义S(a, b, c) = $\sum_{n=0}^{b}F(n)$。

举例来说，如果a = 50、b = 2000以及c = 40，那么F(0) = 3240而F(2000) = 2040。
此外，S(50, 2000, 40) = 5204240。

求S(21⁷, 7²¹, 12⁷)的最后9位数字。

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