A pentagonal number is a figurate number that represents a pentagon. The \( n \)-th pentagonal number can be calculated using the formula: \[ P(n) = \frac{n(3n - 1)}{2} \] where \( n \) is a positive integer. The sequence of pentagonal numbers begins with 1, 5, 12, 22, 35, and so on.
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The th pentagonal number is for a positive integer , giving . A positive integer is pentagonal exactly when is a perfect square and is an integer, by the defining quadratic equation.