Abstract: The integer recursion a_{i+2} = max( a_{i+1},0 ) - a_i defines a sequence of period five for any choice of a_0 and a_1. This is a surprising, arithmetic shadow of the so-called cluster category, which is an object of homological algebra.

The longer recursion a_{i+4} = max( a_{i+3}-a_{i+2}+a_{i+1},0 ) - a_i defines a sequence of period nine for suitable choices of a_0, a_1, a_2, a_3. This is a shadow of a higher cluster category, obtained by switching from three to five terms in the short exact sequences underlying homological algebra.

The talk is a brief tour of these ideas.