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The Thue-Morse Sequence in Base 3/2
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F. M. Dekking

CWI Amsterdam and Delft University of Technology

Faculty EEMCS

P.O. Box 5031

2600 GA Delft

The Netherlands

**Abstract:**

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
*x*_{3/2} of a 2-block substitution on {0, 1}. We prove that
*x*_{3/2} is invariant
for taking the binary complement, and present a list of conjectured
properties of *x*_{3/2},
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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