Number of Distinct Roll Sequences

Encode the recent history

For the common version, a sequence of die rolls must satisfy pairwise gcd and repeat-distance restrictions. The legality of the next face depends on the last one or two faces, so use dp[length][last][previous]: number of valid sequences ending in that ordered pair. Initialize valid one-roll or two-roll states, then append each next face only if every stated rule passes.

The key is to store the minimum history needed for future validation; keeping the full prefix makes the state exponential. With six faces, transitions are constant-sized, so time is O(n * 6³) and space can be O(6²) using two layers. Apply the required modulus at each addition to avoid overflow. Test n = 1, n = 2, a forbidden equal repeat, and a case rejected by the gcd rule. If the prompt's restrictions differ, state them explicitly and adjust the state rather than assuming the standard formulation.