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Solution Sequences for the Keyboard Problem and its Generalizations
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Jonathan T. Rowell

University of North Carolina at Greensboro

Department of Mathematics and Statistics

116 Petty Building

317 College Avenue

Greensboro, NC 27412

USA

**Abstract:**

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.

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(Concerned with sequences
A000792
A131577
A178715
A193286
A193455
A193456
A193457.)

Received June 16 2014;
revised versions received March 18 2015; August 22 2015.
Published in *Journal of Integer Sequences*, October 10 2015.

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