# ThueMorse

ThueMorse[n]

gives the n term in the ThueMorse 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 ThueMorse sequence.
• ThueMorse automatically threads over lists.

# Examples

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## Basic Examples(2)

The fifth element of the ThueMorse 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)

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 ThueMorse sequence can arise from a nested list:

The ThueMorse sequence can arise from the center column of a cellular automaton:

The ThueMorse sequence has a closed form as a hypergeometric function:

Solution to a recurrence relation:

Cube-free sequence:

Overlap-free sequence:

The ThueMorse sequence has a nested structure:

## Neat Examples(1)

Paths at different angles based on ThueMorse 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_2024_thuemorse, author="Wolfram Research", title="{ThueMorse}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/ThueMorse.html}", note=[Accessed: 07-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_thuemorse, organization={Wolfram Research}, title={ThueMorse}, year={2015}, url={https://reference.wolfram.com/language/ref/ThueMorse.html}, note=[Accessed: 07-August-2024 ]}