Thue-Morse Sequence Generator

Take the parity of the number of 1s in the binary form of n, and you have written — in one breath — one of the most surprising words in mathematics: the Thue-Morse sequence, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, ... First studied by Axel Thue in 1906 for its impossible-sounding property — a binary word with no overlapping repetitions — it reappeared in Morse theory, in radar, in chess rules, and in a 19th-century problem of Prouhet about splitting powers perfectly evenly.

Thue-Morse Sequence Generator computes t(n) by the parity definition for any window you choose, annotates each row with the binary form and bit count, and offers the doubling-rule view in the reference tab.

• Parity definition, exactly: t(n) = popcount parity, computed bit by bit for every n.
• Evidence on every row: Binary form and bit count appear next to each value.
• Live balance stats: The 0/1 census and the sliding imbalance of the window.
• Start anywhere: Any index from 0 to 1,000,000, up to 10,000 consecutive terms.
• Reference tab: The doubling rule, overlap-freedom, Prouhet, music and history.
• Free and private: All computation is local; nothing is uploaded, stored or logged.

1. Set the start index — first n to evaluate (0 to 1,000,000).
2. Set the count — consecutive bits to generate (1 to 10,000).
3. Generate — parity of each bit count is shown with the binary evidence.
4. Check the census — balance between 0s and 1s across the window.
5. Copy or download — the run as index = value lines, or as a flat binary word.

Example 1 — The famous opening. Count 16 bits: 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0 — the doubling word 01101001 twice, complemented.

Example 2 — A single bit flips the value. n = 7 (111, three 1s) gives 1, while n = 6 (110, two 1s) gives 0: one less 1 changes the parity completely.

Example 3 — Perfect balance. Count 32: exactly sixteen 0s and sixteen 1s. Every power-of-two window is exactly balanced, a hallmark of the sequence.

Example 4 — No cubes anywhere. Scan count 256: no block of bits appears three times consecutively — the overlap-free property that made Thue's reputation.

Example 5 — Large indices. Start 999,983, count 16: twenty-bit indices still flip signs by popcount parity as the window shifts.

• Definition-faithful: The parity rule decides every bit; the doubling view is derived, not assumed.
• Educational rows: Binary digits and counts make each answer self-explanatory.
• Insightful stats: Balance and census show the deeper equidistribution.
• Flexible ranges: Any window of up to 10,000 terms from any origin.
• Clean export: Copy or download per-index lines or a flat binary word.
• Private by design: No servers, no uploads, no logs.