Journal of Integer Sequences, Vol. 26 (2023), Article 23.2.3

The Thue-Morse Sequence in Base 3/2

F. M. Dekking
CWI Amsterdam and Delft University of Technology
Faculty EEMCS
P.O. Box 5031
2600 GA Delft
The Netherlands


We discuss the base 3/2 representation of the natural numbers. We prove that the sum-of-digits function of the representation is a fixed point of a 2-block substitution on an infinite alphabet, and that this implies that sum-of-digits function modulo 2 of the representation is a fixed point x3/2 of a 2-block substitution on {0, 1}. We prove that x3/2 is invariant for taking the binary complement, and present a list of conjectured properties of x3/2, which we think will be hard to prove. Finally, we make a comparison with a variant of the base 3/2 representation, and give a general result on p-q-block substitutions.

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(Concerned with sequences A000002 A000120 A007953 A010060 A024629 A053735 A244040 A244041 A357448.)

Received February 9 2023; revised version received February 12 2023. Published in Journal of Integer Sequences, February 23 2023.

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