Triangular Number Calculator

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.
The story attached to it is Gauss as a schoolboy, told to add 1 to 100 to keep him busy, and answering 5050 almost at once by pairing 1 with 100, 2 with 99 and so on: fifty pairs of 101. Whether it happened quite like that is anyone’s guess, but the pairing trick is the proof, and T₁₀₀ = 100 × 101 / 2 = 5050.

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

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

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.