Escher\u27s Problem and Numerical Sequences

Counting problems lead naturally to integer sequences. For example if one asks for the number of subsets of an $n$-set, the answer is $2^n$, or the integer sequence $1,~2,~4,~8,~ldots$. Conversely, given an integer sequence, or part of it, one may ask if there is an associated counting problem. There might be several different counting problems that produce the same integer sequence. To illustrate the nature of mathematical research involving integer sequences, we will consider Escher\u27s counting problem and a generalization, as well as counting problems associated with the Catalan numbers, and the Collatz conjecture. We will also discuss the purpose of the On-Line-Encyclopedia of Integer Sequences