Adjacent-Digit Distinctness in Triangular Numbers of 9t-Digit Repdigits

We study OEIS A050759/A050760, where A050759 records the indices \(n\) for which the triangular number \(T_n=n(n+1)/2\) has no pair of equal adjacent decimal digits and A050760 records the corresponding triangular numbers. For \(d∈\{1,…,9\}\) and \(t≥ 1\), let\[nd,t=d{10⁹ᵗ-1}{9},\]the repdigit consisting of \(9t\) copies of \(d\). We derive exact base-\(10^9\) block expansions for every \(T_{nd,t}\) and prove the classification\[T_{nd,t} has no equal adjacent decimal digits∈\{1,2,4,5,8\}.\]Thus each of the five digits \(1,2,4,5,8\) supplies an explicit infinite subfamily of both A050759 and A050760. For \(d=3,6,9\) the obstruction occurs inside a fixed block, while for \(d=7\) it is the fixed central junction \(5|5\). The proof is an exact block decomposition and does not rely on finite computation.