Journal of Integer Sequences, Vol. 17 (2014), Article 14.11.7

On Collatz Words, Sequences, and Trees

Wolfdieter Lang


Motivated by recent work of Trümper, we consider the general Collatz word (up-down pattern) and the sequences following this pattern. We derive recurrences for the first and last sequence entries from repeated application of the general solution of a binary linear inhomogeneous Diophantine equation. We solve these recurrences and also discuss the Collatz tree.

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(Concerned with sequences A000010 A005186 A007583 A016957 A107711 A127824 A152007 A176866 A234038 A238475 A238476 A239125 A239126 A239127 A239130 A240222 A240223.)

Received April 10 2014; revised versions received October 2 2014; November 5 2014; November 11 2014. Published in Journal of Integer Sequences, December 12 2014.

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