Hexagonal Number Calculator
Find nth hexagonal number with n(2n-1) and next value.
What it does
Hexagonal numbers arrange dots in nested hexagons. Each shell adds a hexagonal ring. Totals grow near twice n squared.
The tool above takes n and returns hexagonal, next, and triangular n. The default 100 matches the worked example, so the page opens showing 19,900. Type a new n to update all three results.
The form n(2n-1) needs one double, one shift, and one multiply. For 100 it is 100 times 199. Triangular n(n+1)/2 sits alongside: 100 gives 5,050.
Use this for figurate studies and sequence checks. Hexagonal numbers equal every other triangular: H(n) = T(2n-1).
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
- Pick position n.
- Type n as a whole number.
- Keep n between 1 and 1000000.
- Read Hexagonal as the main result.
- Read Next for n+1.
- Use Triangular n to compare families.
Formula
- n
- Position, whole number.
- H(n)
- nth hexagonal number.
- T(n)
- Triangular n for reference.
Results stay whole for whole n. Cap at 1000000 keeps integers exact.
Worked example
Example: n = 100
Find the 100th hexagonal number.
- Write n = 100.
- 2n-1 = 199.
- Multiply: 100 x 199 = 19,900.
- Triangular 100 = 5,050.
- Next at 101: 101 x 201 = 20,301.
- Check: hex grows near 2n squared.
Reference table
| n | Hexagonal | Triangular |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 6 | 3 |
| 10 | 190 | 55 |
| 100 | 19,900 | 5,050 |
Limits
- Whole n only 1 to 1000000.
- Standard hexagonal here, not centered.
- Large values need commas.
- No partitions computed here.
Common mistakes
- Using n(3n-1)/2 pentagonal form by mistake.
- Doubling after multiplying instead of before.
- Entering n = 0.
- Mixing hexagonal with hexadecimals.
- Reading next as plus n.
Try it yourself
Quick checks. Answers sit tucked under each question.
Hex 10?
190 with next 231 and triangular 55.
Triangular 100?
5,050.
Questions
Hexagonal?
n(2n-1).
100th?
19,900.
Next?
Form at n+1. Here 20,301.
Triangular?
n(n+1)/2. Here 5,050.
Max n?
1000000 here.
Use?
Figurate studies.