Journal of Integer Sequences, Vol. 18 (2015), Article 15.11.6

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

Sam Northshield
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) 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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