Recursion relation and initial conditions Write a recursion relation and initial conditions for the number of words of length n using the letters A,B,C  such that the number of the letter 'A' is even. 
(A occur an even number of times)

$o_{n}=$ number of words having $n$ letters and with odd number of
letter $A$
$e_{n}=$ number of words having $n$ letters and with even number
of letter $A$
Then $o_{0}=0$ and $e_{0}=1$ (empty word counts).
$o_{n+1}=2o_{n}+e_{n}$
$e_{n+1}=o_{n}+2e_{n}$
Then $o_{n}+e_{n}=3^{n}$ leads to $e_{n+1}=3^{n}+e_{n}$
