Star Number Calculator

Centered star counts with 6n(n-1)+1 and points shown.

What it does

Star numbers draw a six pointed star with dots. A center dot anchors twelve spokes that grow each layer. Totals follow six times n times n minus one, plus one.

The tool above takes n and returns star, next, and points. The default 5 matches the worked example, so the page opens showing 121. Type a new n to update all three results.

The form 6n(n-1)+1 needs one shift, one multiply, and one add. For 5 it is 6 times 5 times 4 plus 1 for 121. Points count 12 times n minus 1: 48 here. Next at 6 is 181.

Use this for centered figurate studies alongside hexagonal and triangular families. Stars grow near six n squared.

This page shows the formula, one full hand checked example at n = 5, 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 Star as the main result.
5. Read Next for n+1.
6. Use Points for spoke dots.

Formula

n

Position, whole number.

S(n)

nth star number.

points

Outer dots excluding center.

Center 1 plus points gives star. Cap at 1000000 keeps integers exact.

Worked example

Reference table

Limits

• Whole n only 1 to 1000000.
• Standard stars here.
• Large values need commas.
• No drawings here, counts only.

Common mistakes

• Forgetting plus 1 center.
• Using 12n instead of 12(n-1) for points.
• Entering n = 0.
• Mixing star with pentagonal forms.
• 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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