Rudin-Shapiro Numbers Generator

Count the adjacent "11" pairs in the binary expansion of n, and take a parity vote: even gives +1, odd gives -1. The result is the Rudin-Shapiro sequence — 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, -1, -1, 1, -1, ... — a +1/-1 sequence built to defeat a folklore conjecture of Littlewood, and today a quiet workhorse of polyphase signal design.

Rudin-Shapiro Numbers Generator counts the overlapping 11-pairs exactly for any starting index, renders each value with its binary form and pair count, and tracks the running sum so the famously flat partial sums are visible immediately.

• Exact pair counting: Overlapping windows handled precisely, so "111" contributes two pairs as it should.
• Binary, count and value on every row: See the arithmetic that decides each sign.
• Running sum column: The flat cumulative behavior that made the sequence famous is displayed live.
• Start anywhere: Any index from 0 to 1,000,000, up to 10,000 consecutive terms.
• Balance statistic: The tools show how far the block wanders from equal signs.
• Reference tab: Polynomial story, Golay/Shapiro history, radar application, OEIS links.
• Free and private: All computation is local; nothing is uploaded or logged.

1. Set the start index — first n to evaluate (0 to 1,000,000).
2. Set the count — consecutive values to generate (1 to 10,000).
3. Generate — each row shows binary(n), the 11-pair count, the parity, and r(n).
4. Watch the running sum — it hovers near zero because pairs come in canceling runs.
5. Copy or download — the run as index = value lines.

Example 1 — First 8 values. Count 8 gives +1, +1, +1, -1, +1, +1, -1, +1. The first -1 arrives at n = 3, whose binary form 11 holds exactly one pair.

Example 2 — The overlap rule. n = 7 (binary 111) holds two pairs — "11" twice — an even tally, so r(7) = +1 even though 7 is dense with 1s.

Example 3 — Three runs of 1s. n = 15 (binary 1111) counts three pairs, giving -1. Long runs alternate in sign by their length minus one.

Example 4 — Balance check. Count 16: six of sixteen values are -1, and the running sum at the end is 4 — small random walks, exactly the promised flatness.

Example 5 — Large indices. Start 987,654, count 8: binary strings of 20 digits, pair counts tiny next to the indices — signs stay determined by the last runs.

• Definition-faithful: Overlapping pairs counted exactly, nothing skipped.
• Self-explanatory output: Every sign carries its own evidence on the same row.
• Live balance: The running sum makes the celebrated flatness visible instantly.
• Flexible ranges: Start deep in the sequence and inspect any window up to 10,000 terms.
• Clean export: Copy or download index = value lines for analysis.
• Private by design: No server involved; nothing uploaded, stored or logged.