For $n \geq 1$, let $t_n$ denote the number of elements in $\{0,1,2,3\}^n$ which have the property that no elements is followed immediately by itself (i.e. we don't allow $\{...,0,0,...\}$ or $\{...,2,2,...\}$, for example).
Please let me know the approach below is acceptable:
Let $a_n, b_n, c_n, d_n$ count the strings satisfying the requirement, starting with 0,1,2,3, respectively.
Then we would have: $$ t_n = a_n+b_n+c_n+d_n \\a_n = t_{n-1} - a_{n-1} \\b_n = t_{n-1} - b_{n-1} \\c_n = t_{n-1} - c_{n-1} \\d_n = t_{n-1} - d_{n-1} $$
So $$t_n = 4t_{n-1} - (a_{n-1}+b_{n-1}+c_{n-1}+d_{n-1})= 3t_{n-1}$$
I also have problem setting up the initial condition. With the way the problem is worded, one can easily see that $t_1$ should be 4, and $t_2$ should be 12. But when I plugged these conditions into wolfram separately, the two initial conditions gave two different answers. Specifically, for $t_1=4$, there is no solution, and for $t_2=12$, $t_n = 2 \times 4^{n-1}$. My gut tells me that I should go with $t_2=12$, and define $t_1=4$ as a special case. Am I correct to assume so?
Yes, you are correct to assume so. Those recurrence relations of yours are only valid for $n\ge 2$.
But there is a simpler approach: just observe that the first element of your sequence can take four values, and subsequent elements can take three values. So you get $t_n=4\times 3^{n-1}$.