Editing Thue–Morse sequence (section)

=== Characterization using bitwise negation ===
The Thue–Morse sequence in the form given above, as a sequence of [[bit]]s, can be defined [[recursion|recursively]] using the operation of [[bitwise negation]].
So, the first element is 0.
Then once the first 2<sup>''n''</sup> elements have been specified, forming a string ''s'', then the next 2<sup>''n''</sup> elements must form the bitwise negation of ''s''.
Now we have defined the first 2<sup>''n''+1</sup> elements, and we recurse.

Spelling out the first few steps in detail:
* We start with 0.
* The bitwise negation of 0 is 1.
* Combining these, the first 2 elements are 01.
* The bitwise negation of 01 is 10.
* Combining these, the first 4 elements are 0110.
* The bitwise negation of 0110 is 1001.
* Combining these, the first 8 elements are 01101001.
* And so on.

So
* ''T''<sub>0</sub> = 0.
* ''T''<sub>1</sub> = 01.
* ''T''<sub>2</sub> = 0110.
* ''T''<sub>3</sub> = 01101001.
* ''T''<sub>4</sub> = 0110100110010110.
* ''T''<sub>5</sub> = 01101001100101101001011001101001.
* ''T''<sub>6</sub> = 0110100110010110100101100110100110010110011010010110100110010110.
* And so on.

In [[Python (programming language)|Python]]:
<syntaxhighlight lang="python">
def thue_morse_bits(n):
	"""Return an int containing the first 2**n bits of the Thue-Morse sequence, low-order bit 1st."""
	bits = 0
    for i in range(n):
        bits |= ((1 << (1 << i)) - 1 - bits) << (1 << i)
    return bits
</syntaxhighlight>

Which can then be converted to a (reversed) string as follows: 
<syntaxhighlight lang="python">
n = 7
print(f"{thue_morse_bits(n):0{1<<n}b}")
</syntaxhighlight>