Work out pentagonal number instantly with clear inputs, formula shown and shareable results.
Pentagonal numbers count dots in nested pentagons sharing a corner, giving Pₙ = n(3n − 1)/2. Consecutive terms differ by 3n + 1, so the gaps grow linearly.
Pentagonal number
Pₙ = n(3n − 1) / 2
12 × 35 / 2 = 210, which happens to also be triangular.
The same formula evaluated at 0, 1, −1, 2, −2, … They appear as exponents in Euler's pentagonal number theorem for partitions.