Triangular Number Calculator
Compute the nth triangular number Tₙ = n(n+1)/2 and check whether a given integer is triangular.
Generates the first N values and shows the sequence.
Triangular Numbers
The nth triangular number Tₙ counts how many dots you need to form a triangle with n dots on each side. The sequence starts 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, … and shows up across combinatorics, geometry, and elementary number theory.
Formula
Tₙ = n × (n + 1) / 2
This is also the sum of the first n positive integers — a classroom result famously discovered by Gauss as a child.
Worked Example — n = 10
T₁₀ = 10 × 11 / 2 = 55
There are 55 handshakes among 11 people if each pair shakes once, and 55 dots in a triangle of side 10.
Properties
| Property | Statement |
|---|---|
| Recursion | Tₙ = Tₙ₋₁ + n |
| Connection to squares | Tₙ + Tₙ₋₁ = n² |
| Sum of two | Tₙ + Tₙ₊₁ = (n+1)² |
| Cubes | 1³ + 2³ + … + n³ = Tₙ² |
| Combinatorial | Tₙ = C(n+1, 2) |
Is N a Triangular Number?
A non-negative integer N is triangular if and only if 8N + 1 is a perfect square. If so, n = (√(8N + 1) − 1) / 2.
For example, 91 → 8 × 91 + 1 = 729 = 27², so n = (27 − 1)/2 = 13 → T₁₃ = 91.
Where Triangular Numbers Appear
| Area | Use |
|---|---|
| Combinatorics | Pairs / handshakes from a group |
| Game design | Damage scaling, point pyramids |
| Networking | Number of edges in a complete graph |
| Number theory | Building blocks for figurate-number identities |
| Bowling | Pins in standard 4-row formation = T₄ = 10 |
| Pool | Standard rack = T₅ = 15 balls |
Triangular and Square Numbers
The only numbers that are both triangular and square are 1, 36, 1225, 41616, 1413721, … This sub-sequence comes from solving the Pell-like equation underlying the two figurate forms.
Tetrahedral Cousins
The 3-D analog is the tetrahedral number: Tetₙ = n(n+1)(n+2)/6. It counts cannon balls stacked in a triangular pyramid — and equals the sum of the first n triangular numbers.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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