How many 10-digit sequence are there?

Probability Level 2

Find the number of 10-digit sequences where the difference between any two consecutive digits is 1, using only the digits 1, 2, 3, and 4.

Examples of such 10-digit sequences are 1234321232 and 2121212121.

Bonus: Let $T(n)$ be the number of such $n$ -digit sequences. What is $\lim_{n \to \infty} \frac{T(n+1)}{T(n)}?$