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An Analogue of Stern’s Sequence for ****Z**[√ 2]

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Sam Northshield

Department of Mathematics

SUNY-Plattsburgh

Plattsburgh, NY 12901

USA

**Abstract:**

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.

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(Concerned with sequences
A001834
A002487
A003500
A007814
A007949
A011900
A046090
A062756.)

Received March 17 2015;
revised versions received December 7 2015; December 13 2015.
Published in *Journal of Integer Sequences*, December 15 2015.

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