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 \(d\in\{1,\ldots,9\}\) and \(t\in\mathbb{Z}_{\ge 1}\), let\[n_{d,t}=d\frac{10^{9t}-1}{9},\]the repdigit consisting of \(9t\) copies of \(d\). We derive exact base-\(10^9\) block expansions for every \(T_{n_{d,t}}\) and prove the classification\[T_{n_{d,t}}\text{ has no equal adjacent decimal digits}\quad\Longleftrightarrow\quadd\in\{1,2,4,5,8\}.\]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.