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Hexagonal Number — Definition, Formula & Examples

A hexagonal number is a figurate number that represents the number of dots that can be arranged in a regular hexagonal pattern. The sequence begins 1, 6, 15, 28, 45, 66, ...

The nnth hexagonal number is defined as Hn=n(2n−1)H_n = n(2n - 1) for positive integers nn. Equivalently, hexagonal numbers form a subset of the triangular numbers: every hexagonal number is also the (2n−1)(2n-1)th triangular number.

Key Formula

Hn=n(2n−1)H_n = n(2n - 1)
Where:
  • HnH_n = The nth hexagonal number
  • nn = A positive integer (the index in the sequence)

How It Works

To find the nnth hexagonal number, substitute nn into the formula Hn=n(2n−1)H_n = n(2n - 1). You can also build the sequence by noting that consecutive hexagonal numbers differ by 4n−34n - 3, since Hn−Hn−1=4n−3H_n - H_{n-1} = 4n - 3. To test whether a given positive integer mm is hexagonal, check whether n=1+1+8m4n = \frac{1 + \sqrt{1 + 8m}}{4} is a positive integer.

Worked Example

Problem: Find the 5th hexagonal number.
Apply the formula: Substitute n=5n = 5 into Hn=n(2n−1)H_n = n(2n - 1).
H5=5(2⋅5−1)H_5 = 5(2 \cdot 5 - 1)
Simplify: Compute the expression inside the parentheses first, then multiply.
H5=5(10−1)=5×9=45H_5 = 5(10 - 1) = 5 \times 9 = 45
Answer: The 5th hexagonal number is 45.

Visualization

Why It Matters

Hexagonal numbers appear in number theory when studying which integers can be expressed as figurate numbers. They connect to triangular numbers (every hexagonal number is triangular) and arise in problems about partitions and polygonal number representations.

Common Mistakes

Mistake: Confusing hexagonal numbers with the count of hexagonal cells (like a honeycomb tiling).
Correction: Hexagonal numbers count dots arranged in nested hexagonal layers around a central dot, not the number of hexagonal tiles. The formula n(2n−1)n(2n-1) gives dot counts, not cell counts.