Recurrence relations: How many numbers between 1 and 10,000,000 don't have the string 12 or 21 so the question is (to be solved with recurrence relations: How many numbers between 1 and 10,000,000 don't have the string 12 or 21?
So my solution: $a_n=10a_{n-1}-2a_{n-2}$. The $10a_{n-1}$ represents the number of strings of n length of digits from 0 to 9, and the $2a_{n-2}$ represent the strings of n-length with the 12 or 21 strings included. 
Just wanted to know if my recursion is correct, if so, I'll be able to solve the rest.
Thanks in advance! 

We look at a slightly different problem, from which your question can be answered. 
Call a digit string good if it does not have $12$ or $21$ in it. Let $a_n$ be the number of good strings of length $n$. Let $b_n$ be the number of good strings of length $n$ that end with a $1$ or a $2$,  Then $a_n-b_n$ is the number of good strings of length $n$ that don't end with $1$ or $2$.
We have
$$a_{n+1}=10(a_n-b_n) +9b_n.$$
For a good string of length $n+1$ is obtained by appending any digit to a good string that doesn't end with $1$ or $2$, or by appending any digit except the forbidden one to a good string that ends in $1$ or $2$.
We also have 
$$b_{n+1}=2(a_n-b_n) + b_n.$$ 
For we obtain a good string of length $n+1$ that ends in $1$ or $2$ by appending $1$ or $2$ to a string that doesn't end with either, or by taking a string that ends with $1$ (respectively, $2$) and adding a $1$ (respectively, $2$).  
The two recurrences simplify to 
$$a_{n+1}=10a_n-b_n\qquad\text{ and}\qquad b_{n+1}=2a_n-b_n.$$
For calculational purposes, these are good enough. We do not really need a recurrence for the $a_i$ alone. However, your question perhaps asks about  the $a_i$, so we eliminate the $b$'s.
One standard way to do this is to increment $n$ in the first recurrence, and obtain
$$a_{n+2}=10a_{n+1}-b_{n+1}.$$
But $b_{n+1}=2a_n-b_n$, so 
$$a_{n+2}=10a_{n+1}-2a_n+b_n.$$
But $b_n=10a_n-a_{n+1}$, and therefore
$$a_{n+2}=9a_{n+1}+8a_n.$$ 
Remark: It would have been better to have $b_n$ as above, and $c_n$ the number of strings that do not end in $1$ or $2$, and to forget about $a_n$ entirely for a while. 
