Pentagonal number ( $P_{n}$ )

The

n

P_{n} = n (3 n - 1) /2

n

1, 5, 12, 22, \dots

x

is pentagonal exactly when

24 x + 1

(1 + 24 x + 1) /6

is an integer, by the defining quadratic equation.

Generalized pentagonal number

k (3 k - 1) /2

with

k

any integer, including negative values and zero. For

k = \pm 1, \pm 2, \dots

, positive values occur as pairs

j (3 j - 1) /2

and

j (3 j + 1) /2

. These are the indices in the Euler pentagonal identity.

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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.