Summary: The keyboard problem is an optimization problem asking how many characters can be placed into a blank document using $N$ keystrokes. The question is representative of a larger class of output maximization problems where there is the opportunity to expand output capacity by replicating the existing output as a single unit. Here I define a generalized keyboard sequence as an integer sequence representing the maximum output of such problems, explain the construction of optimal strings of operations leading to these outputs, and demonstrate that each sequence is linearly recursive for sufficiently large $N$. I then evaluate two competing solutions to the keyboard problem and connect additional integer sequences to this class. The article concludes with a brief overview of the crowd-sourcing involved in the keyboard problems initial solution.

11N64, 11N37, 00A08, 11K65, 11B99

keyboard problem, generalized keyboard sequence, doubling sequence, output optimization