Journal of Integer Sequences, Vol. 25 (2022), Article 22.2.1

Self-Containing Sequences, Fractal Sequences, Selection Functions, and Parasequences

Clark Kimberling
Department of Mathematics
University of Evansville
1800 Lincoln Avenue
Evansville, IN 47722


This paper surveys various kinds of ordered sets, with numerous citations to sequences in the On-Line Encyclopedia of Integer Sequences. These ordered sets include self-containing sequences, infinitive sequences, fractal sequences, and parasequences (which are introduced here as a certain type of doubly infinite sequence). Relationships among these are presented, and among more than thirty examples, the Cantor fractal sequence and the Farey fractal sequence are presented. There are several conjectures involving parasequences.

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(Concerned with sequences A000027 A000032 A000045 A000201 A000984 A001045 A001511 A001519 A001622 A001906 A002260 A003602 A003603 A004526 A004736 A004741 A006259 A007336 A007677 A008346 A015518 A019446 A019587 A022446 A022447 A023115 A023133 A029578 A049474 A054065 A054072 A054073 A054108 A054582 A055265 A083582 A084222 A087467 A088370 A098600 A108712 A119015 A120873 A120874 A122196 A125158 A131966 A131967 A131968 A131987 A132223 A132224 A132226 A132283 A132284 A135764 A194959 A279933 A279934 A283939 A349554.)

Received July 20 2021; revised version received January 6 2022. Published in Journal of Integer Sequences, January 28 2022.

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