A string forms an addition equation if it consists of
- Digits, forming positive integers. No leading zeros are allowed.
- A single plus sign between integers.
- A single equality sign.
Most importantly, the equality must hold.
Example of strings of length 7 forming addition equations:
1+1+1=3
100=100
77=7+70
1+2=2+1
Note that strings are considered different even if they form equivalent equations, so for example here are three unique strings:
1+2=3
2+1=3
3=1+2
The following strings do not form valid addition equations:
1+1=3
1+1=02
0+1=1
+1=1
2-1=1
Let $A(n)$ be the number of strings of length not larger than $n$ forming addition equations.
You are given $A(3) = 9$, $A(5)=171$ and $A(7)=4878$.
Find $A(50)$. Give your answer modulo $10^9 + 7$.
Solution
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solutions/s990.md.