Counting all possibilities that contain a substring

How many strings are there of seven lowercase letters that have the substring tr in them?

So I am having a little problem with this question, I know that the total number of combinations is $26^6$ but there is double counting on some of the combinations.

For example, when you have a case that contains multiples 'tr' then it will be counted multiple times depending on the location of tr, even though its the same string.

t r t r t r a

Any advice on what to subtract to remove this double counting?

Thanks for any help!

Let $a_n$ be the number of strings of length $n$ with no tr's. An $(n-1)$-long string can be extended in $26$ ways unless it ends in "t", in which case it can only be extended in $25$ ways. So $$a_n= 26a_{n-1}-a_{n-2}$$ for $n\ge2$; the initial conditions are $a_0=1$ and $a_1=26$.

You can give a formula for $a_n$, but to compute $a_7$ it suffices to run out the recurrence. The final answer is $26^7-a_7 = 71,112,600$.

In case there's doubt, another way to verify the recurrence is to write down the $26\times26$ transition matrix for letters. All of the entries are $1$'s except for a single $0$ off the diagonal (corresponding to the prohibited tr). The characteristic polynomial is computed to be $z^{24}(z^2-26z+1)$.