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The Ackermann function

Problem 282 Published on Friday, 12th March 2010, 05:00 pm; Solved by 778;
Difficulty rating: 70%

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For non-negative integers $m$, $n$, the Ackermann function $A(m,n)$ is defined as follows: $$ A(m,n) = \cases{ n+1 &$\htmltext{ if }m=0$\cr A(m-1,1) &$\htmltext{ if }m>0 \htmltext{ and } n=0$\cr A(m-1,A(m,n-1)) &$\htmltext{ if }m>0 \htmltext{ and } n>0$\cr }$$

For example $A(1,0) = 2$, $A(2,2) = 7$ and $A(3,4) = 125$.

Find $\displaystyle\sum_{n=0}^6 A(n,n)$ and give your answer mod $14^8$.