An Analogue of Sterna??s Sequence for Z[sqrt(2)]

摘要

We introduce a sequence b(n) of algebraic integers that is an analogue of Stern's diatomic sequence, not only in definition, but also in many of its properties. Just as Stern's sequence arises from Ford circles, so too b(n) arises from an array of circles. We study the generating function for b(n) and derive several closed formulas for the sequence. Two second order recurrence formulas for b(n) are found. It is shown that, for t the square root of 2, the ratios tb(n+1)/b(n) enumerate the positive rational numbers. Finally, we use b(n) to create a function f(x) that is an analogue of Conway's box function and that has inverse a singular function analogous to Minkowski's question-mark function.