Star Number — Definition, Formula & Examples

A star number is a figurate number that represents a centered six-pointed star (Star of David shape) made of dots. The first few star numbers are 1, 13, 37, 73, 121, and 181.

The $n$ th star number $S_n$ is defined as the number of dots in a centered hexagram pattern with $n$ layers, given by $S_n = 6n(n-1) + 1$ for positive integers $n \geq 1$ .

Key Formula

S_n = 6n(n - 1) + 1

Where:

$S_n$ = The nth star number
$n$ = The layer number (positive integer, starting at 1)

How It Works

To build a star number pattern, start with a single central dot. Each new layer adds a six-pointed star outline around the previous pattern. The first layer adds 12 dots, the second adds 24 dots, and so on — each new layer adds 12 more dots than the one before. You can find any star number directly using the formula rather than adding layer by layer.

Worked Example

Problem: Find the 4th star number.

Write the formula: Use the star number formula with n = 4.

S_4 = 6(4)(4 - 1) + 1

Simplify inside the parentheses: Compute 4 − 1 = 3, then multiply.

S_4 = 6 \times 4 \times 3 + 1 = 72 + 1

Final result: Add 1 to get the answer.

S_4 = 73

Answer: The 4th star number is 73.

Visualization

Why It Matters

Star numbers appear in recreational mathematics and number theory competitions. Checking whether a large number is a star number connects to solving quadratic equations and working with modular arithmetic, since every star number leaves a remainder of 1 when divided by 6.

Common Mistakes

Mistake: Confusing star numbers with centered hexagonal numbers.

Correction: A centered hexagonal number uses a regular hexagon pattern (formula: 3n² − 3n + 1), while a star number uses a six-pointed star pattern (formula: 6n² − 6n + 1). The shapes and formulas are different.