Triangular Numbers Calculator

A triangular number is the number of dots that can be arranged in an equilateral triangle, with one dot in the top row, two in the next, three in the next, and so on. The nth triangular number is the sum of all whole numbers from 1 to n: T(n) = 1 + 2 + 3 + ... + n = n(n+1)/2. The first few are 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55.

T(n) = n(n+1)/2. For example, T(10) = 10 × 11 / 2 = 55 and T(100) = 100 × 101 / 2 = 5,050. This closed-form formula, attributed to the young Carl Friedrich Gauss, avoids adding every integer one by one and works instantly for any size of n.

A whole number x is triangular if and only if 8x + 1 is a perfect square. Compute n = (−1 + √(8x + 1)) / 2; if n is a non-negative whole number, x is triangular and equals T(n). For example, x = 55: 8(55) + 1 = 441 = 21², so n = (−1 + 21)/2 = 10, confirming T(10) = 55.

No. 8(56) + 1 = 449, and √449 ≈ 21.19 is not a whole number, so 56 is not triangular. It falls between T(10) = 55 and T(11) = 66, which are the nearest triangular numbers on either side.

As a schoolboy, Carl Friedrich Gauss is said to have quickly summed 1 through 100 by pairing the first and last terms: (1+100) + (2+99) + ... = 50 pairs of 101 = 5,050. This is exactly T(100), and the pairing trick is the intuitive proof behind the formula n(n+1)/2: pair up n terms into n/2 pairs, each summing to n+1.

T(n) equals the binomial coefficient C(n+1, 2), the number of ways to choose 2 items from a set of n+1. This is because choosing 2 items from n+1 is equivalent to choosing an unordered pair, and there are exactly n(n+1)/2 such pairs. This connection makes triangular numbers appear throughout combinatorics, such as counting handshakes at a party of n+1 people.

T(n) + T(n−1) always equals a perfect square: n². For example, T(5) + T(4) = 15 + 10 = 25 = 5². Geometrically, two consecutive triangles of dots fit together to form an n-by-n square, which is a classic visual proof used in number theory textbooks.

Triangular numbers are the simplest case of polygonal numbers, a family that also includes square numbers (n²), pentagonal numbers (n(3n−1)/2), and hexagonal numbers (n(2n−1)). Each family counts dots arranged in a regular polygon shape. Triangular numbers use the smallest polygon (3 sides), so they grow the slowest among all polygonal number families with more than 2 sides.

Triangular numbers count handshakes among n+1 people, round-robin tournament games (T(n−1) matches for n teams), stacked objects like bowling pins (T(4)=10) or cannonballs, and the number of diagonals plus sides in certain polygon problems. They also appear in computer science when counting pairwise comparisons in nested loops, since a double loop comparing every pair runs in T(n−1) steps.

The standard definition only covers n ≥ 0, giving T(0) = 0 as the smallest triangular number. Extending the formula n(n+1)/2 to negative n produces the same sequence of positive values in reverse order (T(−1) = 0, T(−2) = 1, T(−3) = 3, ...), a curiosity sometimes used in generating-function proofs, but it has no standard combinatorial meaning.

Numbers that are both triangular and square are called square triangular numbers: 0, 1, 36, 1225, 41616, 1413721, ... They grow rapidly and are generated by a Pell equation. The next one after 1 is T(8) = 36 = 6², and after that T(49) = 1225 = 35². These are a well-studied sequence in number theory (OEIS A001110).

Triangular numbers grow quadratically: T(n) ≈ n²/2 for large n. This means T(n) roughly doubles when n increases by about 41% (since (n·√2)² ≈ 2n²). By T(1000) = 500,500 and T(1,000,000) = 500,000,500,000, the values quickly become very large even though n grows only linearly.