An Analogue of Stern’s Sequence for Z[√ 2]
Department of Mathematics
Plattsburgh, NY 12901
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)
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.
Full version: pdf,
(Concerned with sequences
Received March 17 2015;
revised versions received December 7 2015; December 13 2015.
Published in Journal of Integer Sequences, December 15 2015.
Journal of Integer Sequences home page