Adjacent-Digit Distinctness in Squares of Generalized Repunits

For an integer base \(b\ge 3\), let \[ R_n^{(b)}=1+b+\cdots+b^{n-1}=\frac{b^n-1}{b-1} \] be the length-\(n\) repunit in base \(b\). We prove that the standard base-\(b\) expansion of \(\bigl(R_n^{(b)}\bigr)^2\) has no two equal adjacent digits if and only if \(n=1\) or \(n\not\equiv 1\pmod{b-1}\). The proof follows the carries generated by the triangular convolution coefficients of the square and shows that the only possible equality of adjacent digits occurs at the central transition. Specializing to \(b=10\) gives an exact criterion for decimal repunit squares. In particular, the infinite family \[ \left(\frac{10^{9t}-1}{9}\right)^2,\qquad t\ge 1, \] previously exhibited by Rodrigo is recovered as a special case, linking the classification to OEIS A090516.Version 2: Added attribution and a reference to Rodrigo's 2017 construction of the decimal repunit-square family used in the A090516 specialization, and revised the corresponding historical positioning in the abstract, introduction, decimal specialization, and conclusion. The arbitrary-base classification theorem and its proof are unchanged.