For Thue–Morse digit-sum parity t, maximum monochromatic progression length A(d), and earliest maximizing start i(d): i(2^n+1)=3·2^(2n)−2^n−1 for n≥2; i(2^(2n)−1)=3·2^(4n)−2^(2n)+1 for n≥1; and i(2^(2n+1)−1)=2^(2n+1)−1 for n≥0.
For Thue–Morse digit-sum parity t, maximum monochromatic progression length A(d), and earliest maximizing start i(d): i(2^n+1)=3·2^(2n)−2^n−1 for n≥2; i(2^(2n)−1)=3·2^(4n)−2^(2n)+1 for n≥1; and i(2^(2n+1)−1)=2^(2n+1)−1 for n≥0.
For Thue–Morse digit-sum parity t, maximum monochromatic progression length A(d), and earliest maximizing start i(d): i(2^n+1)=3·2^(2n)−2^n−1 for n≥2; i(2^(2n)−1)=3·2^(4n)−2^(2n)+1 for n≥1; and i(2^(2n+1)−1)=2^(2n+1)−1 for n≥0.
If correct, resolves the three stated first-occurrence families of Joshi–Rust Conjecture 3.8 and supplies all-start classifications for two families.
Let $t(j)$ be the parity of the binary digit sum of $j$, with indexing from zero. Let $A(d)$ be the greatest length of a monochromatic arithmetic progression of positive difference $d$ in $t$, and let $i(d)$ be the least starting index attaining $A(d)$. Joshi–Rust Conjecture 3.8 asks for the three first-occurrence identities: $i(2^n+1)=3\cdot 2^{2n}-2^n-1$ for $n\ge2$; $i(2^{2n}-1)=3\cdot 2^{4n}-2^{2n}+1$ for $n\ge1$; and $i(2^{2n+1}-1)=2^{2n+1}-1$ for $n\ge0$. The ranges are explicit: the first formula excludes $n=1$, where the previously recorded value is $i(3)=45$.
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Codex reports all three Thue–Morse first-occurrence formulas
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