Pentagonal Number Calculator

Find nth pentagonal number with n(3n-1)/2 and next value.

What it does

Pentagonal numbers arrange dots in nested pentagons. Each layer adds a pentagonal ring around the last. Totals grow near 1.5 times n squared.

The tool above takes n and returns pentagonal, next, and hexagonal n. The default 100 matches the worked example, so the page opens showing 14,950. Type a new n to update all three results.

The form n(3n-1)/2 needs one multiply, one shift, and one halve. For 100 it is 100 times 299 over 2. Hexagonal n(2n-1) sits alongside for comparison: 100 gives 19,900.

Use this for figurate studies, partitions where pentagonal theorem appears, and sequence checks. Euler used generalized pentagonals in partition recurrences.

This page shows the formula, one full hand checked example at n = 100, a short table, and practice checks that run through the tool.

How to use

1. Pick position n.
2. Type n as a whole number.
3. Keep n between 1 and 1000000.
4. Read Pentagonal as the main result.
5. Read Next for n+1.
6. Use Hexagonal n to compare families.

Formula

n

Position, whole number.

P(n)

nth pentagonal number.

H(n)

nth hexagonal number.

3n-1 stays odd or even to keep numerator even, so results stay whole. Cap at 1000000 keeps integers exact.

Worked example

Reference table

Limits

• Whole n only 1 to 1000000.
• Centered versus standard pentagonals differ. This is standard.
• Large values need commas.
• No partitions computed here.

Common mistakes

• Using 3n+1 instead of 3n-1.
• Halving before multiplying and losing halves.
• Mixing pentagonal with hexagonal forms.
• Entering n = 0.
• Reading next as plus n.

Try it yourself

Quick checks. Answers sit tucked under each question.

Questions

Sources

• OEIS: Triangular numbers
• NIST DLMF: Combinatorial Analysis

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