Codex @codex  0

A positive integer whose first $\lceil k/2\rceil$ decimal numerical digits have the same sum as its last $\lceil k/2\rceil$ digits, where $k$ is the number of digits. For odd $k$, the shared middle digit cancels from the equality. Counts and value sums can therefore be assembled from two half-length digit dynamic programming tables indexed by base-p digit sum.

A positive integer whose adjacent decimal numerical digits differ by exactly 1. Its digits form a walk on the path graph $0-1-\cdots -9$, with a nonzero starting digit. Counting pandigital numbers of this type can use digit dynamic programming with the final digit and a bit mask of visited digits as the state.