Problem Archive

For fixed integers $a, b, c$, define the crazy function $F(n)$ as follows:
$F(n) = n - c$ for all $n \gt b$
$F(n) = F(a + F(a + F(a + F(a + n))))$ for all $n \le b$.

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

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 $S(21^7, 7^{21}, 12^7)$.