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Pentagonal Number Calculator

Work out pentagonal number instantly with clear inputs, formula shown and shareable results.

About Pentagonal Number Calculator

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.

Formula

Pentagonal number

Pₙ = n(3n − 1) / 2

Worked example

With the default values below, Pentagonal Number Calculator returns nth pentagonal number of 210. Change any input above and every figure here is recalculated in your browser.

Inputs used

Index naccepts 1–10,000,000
12
How many terms to listaccepts 1–20
6

Results

nth pentagonal number
210
Next pentagonal number
247
Gap to the next term
37
First terms of the sequence
1, 5, 12, 22, 35, 51

Step by step

  1. 1Values used
  2. 2Pentagonal number
  3. 3nth pentagonal number= 210
  4. 4Next pentagonal number= 247
  5. 5Gap to the next term= 37
  6. 6First terms of the sequence= 1, 5, 12, 22, 35, 51

Assumptions and limits

  • The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is the 12th pentagonal number?

12 × 35 / 2 = 210, which happens to also be triangular.

What are generalised pentagonal numbers?

The same formula evaluated at 0, 1, −1, 2, −2, … They appear as exponents in Euler's pentagonal number theorem for partitions.

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