Summary: We investigate the recursive structure of differentiable sequences over the alphabet ${1, 2}$. We derive a recursive formula for the $(n + 1)$-th symbol of a differentiable sequence, which yields to a new recursive formula for the Kolakoski sequence. Finally, we show that the sequence of absolute differences of consecutive symbols of a differentiable sequence $u$ is a morphic image of the run-length encoding of $u$.

68R15, 11Y55, 11B83

kolakoski sequence, integer sequences, differentiable sequences, smooth sequences, combinatorics of words