We give an explicit formula for OEIS sequence A265233 as a binomial sum involving the Franel numbers. The sequence counts three-row arrays that contain equally many copies of each of three symbols and have no equal vertically adjacent entries. Reading the array row by row, we label the symbols in order of first occurrence. A decomposition of the column-counting polynomial yields the explicit formula and a proof of Mathar's conjectured recurrence. We then derive the hypergeometric generating function recorded by van Hoeij and the asymptotic formula conjectured by Kotesovec. The latter follows from a logarithmic singular expansion with an explicit error estimate.