Triangular numbers

Published juin 20, 2026 · By NumberWiki

A triangular number counts the dots you need to build a filled triangle: 1, 3, 6, 10, 15, 21, 28, … Add up the whole numbers from 1 to k and you get the k-th triangular number. They are the friendliest of the figurate numbers, and they hide a surprising amount of mathematics.

Stacking dots into triangles

Picture bowling pins or a rack of billiard balls. One dot is the first triangular number. Add a row of two beneath it: three dots (3). Add a row of three: six (6). Add a row of four: ten (10) — the bowling rack. Each new row adds one more dot than the last, so the k-th triangular number is the running total of 1 + 2 + 3 + ⋯ + k:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, …

Gauss and the shortcut

There is a closed form: the k-th triangular number is T_k = k(k+1)/2. The story usually told to explain it involves a young Carl Friedrich Gauss. As the tale goes, his schoolteacher — hoping for a few minutes' peace — told the class to add up all the whole numbers from 1 to 100. Gauss produced the answer almost at once: 5,050.

His trick is the proof of the formula. Write the sum forwards and backwards and add the columns:

1 + 2 + 3 + … + 98 + 99 + 100
100 + 99 + 98 + … + 3 + 2 + 1

Every column sums to 101, and there are 100 columns, giving 100 × 101 = 10,100 — but that double-counts, so the real total is half of it: 5,050 = 100 · 101 / 2. The same argument with k in place of 100 gives T_k = k(k+1)/2. (Whether the schoolroom anecdote is literally true is doubted by historians, but it captures a genuine and beautiful idea.)

Two triangles make a square

Triangular numbers are woven into the rest of arithmetic. The cleanest identity: two consecutive triangular numbers add to a perfect square. T_k−1 + T_k = k². For example T₃ + T₄ = 6 + 10 = 16 = 4². Geometrically, two triangular arrangements slot together into a square grid.

A few more appearances:

The handshake count. In a room of k+1 people, the number of distinct handshakes is T_k — which is why the triangular numbers are also the binomial coefficients $\binom{k+1}{2}$, counting ways to choose 2 from k+1.
Every even perfect number is triangular. 6, 28, 496, 8128 are all triangular — a consequence of the Euclid–Euler form 2^p−1(2^p−1).
Hexagonal numbers are a subset. Every hexagonal number is triangular (the odd-indexed ones), so the two families overlap richly.
The doubles are the pronic numbers. Twice a triangular number, k(k+1), is a pronic (oblong) number.

Gauss's Eureka theorem

Triangular numbers anchor one of the prettiest results in number theory. In 1796 Gauss recorded in his diary the single triumphant line — "ΕΥΡΗΚΑ! num = Δ + Δ + Δ" — on proving that every positive integer is the sum of at most three triangular numbers. (The Δ stands for a triangular number.) For instance 50 = 1 + 21 + 28 and 100 = made from three triangulars too. It is a special case of Fermat's polygonal number theorem, which says every integer is a sum of at most m m-gonal numbers.

Curiosities

The only Fibonacci numbers that are also triangular are 1, 3, 21, and 55 — a finite list.
Square triangular numbers — both a square and a triangle — are infinite but rare: 1, 36, 1225, 41616, 1413721, … generated from the Pell numbers.
T₃₆ = 666, the "number of the beast," is the sum of 1 through 36 — and 36 is itself both square and triangular.
The digital roots of the triangular numbers cycle with period nine: 1, 3, 6, 1, 6, 3, 1, 9, 9, then repeat.