A square triangular number is a number that is both a perfect square and a triangular number. A triangular number is a number that can be arranged in the shape of an equilateral triangle. The \(n\)-th triangular number is given by the formula: \[ T_n = \frac{n(n + 1)}{2} \] where \(n\) is a positive integer. A perfect square is a number that can be expressed as the square of an integer.
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A square triangular number is simultaneously a triangular number and a perfect square. For example, . This differs from a squared triangular number: is not itself triangular, by the recognition criterion for triangular numbers. Writing gives , a Pell equation that explains why this intersection of two counting patterns has additional arithmetic structure.