Triangular Numbers Calculator
Generate triangular numbers T(n) = n(n+1)/2, view dot-pattern visualization, and see step-by-step formula breakdown for any positive integer n.
What Are Triangular Numbers?
Triangular numbers count dots arranged in an equilateral triangle. The $n$th triangular number is:
$$T_n = \frac{n(n+1)}{2} = 1 + 2 + 3 + \cdots + n$$
First Few Triangular Numbers
$T_0 = 0$, $T_1 = 1$, $T_2 = 3$, $T_3 = 6$, $T_4 = 10$, $T_5 = 15$, $T_6 = 21$, $T_7 = 28$.
Applications
Triangular numbers appear in combinatorics, network connections (a complete graph with $n$ nodes has $T_{n-1}$ edges), and the handshake problem.
Related: Sum of Linear Number Sequence Calculator, Pascal Triangle Generator, and Sum of Series Calculator.
Frequently Asked Questions
What is T(10)?
T(10) = 10 × 11 / 2 = 55.
Is 1 a triangular number?
Yes. T(1) = 1. It is the first non-zero triangular number.
How do adjacent triangular numbers relate to squares?
Two consecutive triangular numbers sum to a perfect square: T(n) + T(n+1) = (n+1)².
What is T(0)?
T(0) = 0. An empty triangle has zero dots.
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