Summary: We introduce a generalization of the Conway-Hofstadter $10,000 sequence. The sequences introduced, called $k-sequences$, preserve the Conway-Hofstadter-Fibonacci-like structure of forming terms in the sequence by adding together two previous terms, equidistant from the start and end of the sequence. We examine some particular $k$-sequences, investigate relationships to known integer sequences, establish some properties which hold for all $k$, and show how to solve many of the defining nonlinear recursions by examining related underlying sequences termed $clock$ sequences.$

conway-hofstadter, Fibonacci sequence, nonlinear recursion