An insect is situated on the vertex of a regular tetrahedron.

The problem can be interpreted as follows:

To find the number of strings of length $7$ that can be made from the letters $A , B , C , D$ such that the first and the last letters are both $A$ and there are no two consecutive letters in any position.

We can solve this by considering cases on the first time the letter $A$ occurs after the first place.Then we get the recurrence ,

$f(6) = 3\times f(4) + 3\times 2 f(3) + 3 \times 2^2 f(2) + 3 \times 2^4$

It is simple to see that $f(2) = 3 , f(3) = 6$ and $f(4) = 21$ .

We then get $\boxed{f(6) = 183}$