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ThueMorse[n]

gives the n^(th) term in the Thue–Morse sequence.

Details
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Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
Neat Examples  
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ThueMorse

ThueMorse[n]

gives the n^(th) term in the Thue–Morse sequence.

Details

  • ThueMorse[n] is 1 exactly when n has an odd number of 1s in its binary expansion.
  • ThueMorse gives the sequence generated by the substitution starting from .
  • ThueMorse[0] is the beginning of the Thue–Morse sequence.
  • ThueMorse automatically threads over lists.

Examples

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Basic Examples  (2)

The fifth element of the Thue–Morse sequence:

Five has an even number of ones in its binary expansion:

The first 10 elements of the sequence:

Display the values alongside the binary expansion:

Scope  (2)

ThueMorse threads over lists:

Evaluate at large integers:

Applications  (1)

The Koch snowflake:

Properties & Relations  (8)

ThueMorse[n] is 1 if and only if n has an odd number of 1s in its binary form:

The Thue–Morse sequence can arise from a nested list:

The Thue–Morse sequence can arise from the center column of a cellular automaton:

The Thue–Morse sequence has a closed form as a hypergeometric function:

Solution to a recurrence relation:

Cube-free sequence:

Overlap-free sequence:

The Thue–Morse sequence has a nested structure:

Neat Examples  (1)

Paths at different angles based on Thue–Morse sequences:

Wolfram Research (2015), ThueMorse, Wolfram Language function, https://reference.wolfram.com/language/ref/ThueMorse.html.

Text

Wolfram Research (2015), ThueMorse, Wolfram Language function, https://reference.wolfram.com/language/ref/ThueMorse.html.

CMS

Wolfram Language. 2015. "ThueMorse." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ThueMorse.html.

APA

Wolfram Language. (2015). ThueMorse. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ThueMorse.html

BibTeX

@misc{reference.wolfram_2025_thuemorse, author="Wolfram Research", title="{ThueMorse}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/ThueMorse.html}", note=[Accessed: 15-August-2025]}

BibLaTeX

@online{reference.wolfram_2025_thuemorse, organization={Wolfram Research}, title={ThueMorse}, year={2015}, url={https://reference.wolfram.com/language/ref/ThueMorse.html}, note=[Accessed: 15-August-2025]}