Only 0, 1, 2, 3, 4

Call the numbers with 0, 1, 2, 3, 4 with difference between consecutive digits 1, as good numbers.

Let $A_n$ denote the number of good n-digit sequences that end in either 0 or 4; let $B_n$ denote the number of good n-digit sequences that end in either 1 or 3; and let $C_n$ denote the number of good n-digit sequences that end in 2. Then for non-negative integers n:

(a) $A_{n+1}$ = $B_n$ , because each sequence in $A_{n+1}$ can be converted into a sequence in $B_n$ by deleting its last digit.

(b) $B_{n+1}$ = $A_n+2C_n$ , because each sequence in $A_n$ can be converted into a sequence in $B_{n+1}$ by adding a 1 (if it ends with 0) or a 3 (if it ends with 4) to its end, and each sequence in $C_n$ can be converted into a sequence in $B_{n+1}$ by adding a 1 or a 3 to it.

(c) $C_{n+1}$ = $B_n$ , because each sequence in $C_{n+1}$ can be converted into a sequence in $B_n$ by deleting the 2 at its end.

Hence $B_{n+1}$ = $3B_{n-1}$ for n>1. It is easy to see that $B_1$ = 2 and $B_2$ = 4. Hence $B_{2n+1}$ = $2.3^n$ and $B_{2n}$ = $4.3^{n-1}$ .

It follows that there are $A_10+B_10+C_10 = 2B_9 + B_10 = 4.3^4+4.3^4 = 648$ good ten-digit sequences. .