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The Thue-Morse sequence is defined as a binary sequence and can be generated like

0, 01, 01 10, 01 10 10 01, 01 10 10 01 10 01 01 10, ... .

So the second half of the series is always the binary complement of the first half of the series.

But is there a way to generate an analogous ternary sequence? Intuitively my first guess for a ternary Thue-Morse sequence was like

0, 01, 012, 012 120, 012 120 120 201, 012 120 120 201 120 201 201 012, ...

So here the second half of the series is the "ternary complement" (rotation 0->1, 1->2, 2->0 instead of 0->1, 1->0) of the first half.

But it could also be

0, 01, 012, 012 120 201, 012 120 201 120 201 012 201 012 120, ...

Here the second third is the "ternary complement" of the first third and the third third is the "ternary complement" of the second third.

Does any of my constructions for a ternary Thue-Morse series make sense? Is there maybe a unique way to generate an analogous ternary sequence? And how to construct n-ary versions of the Thue-Morse series in general?

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