The triangular numbers are \(T_k = k(k+1)/2\) — the number of dots needed to form an equilateral triangle of side \(k\). The first few: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66.
Every integer can be written as the sum of at most three triangular numbers (Gauss's Eureka theorem of 1796 — "ΕΥΡΗΚΑ! num = Δ + Δ + Δ"). The sum of two consecutive triangular numbers is always a perfect square: \(T_{k-1} + T_k = k^2\).
Triangular numbers count many things: handshakes among \(k+1\) people, edges in a complete graph \(K_{k+1}\), the moves in the Tower of Hanoi for \(k\) disks (well, \(2^k - 1\), but the recursive structure relates), and the entries in the lower triangle of an \((k+1) \times (k+1)\) matrix.