A pronic (or oblong or rectangular) number is a number of the form \(k(k+1)\) for some non-negative integer \(k\) — the product of two consecutive integers.
The first pronic numbers: 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156.
Geometrically, a pronic number counts the dots in a \(k \times (k+1)\) rectangle. Pronic numbers are exactly twice the triangular numbers: \(k(k+1) = 2 T_k\).
The sum of the first \(k\) even numbers is \(k(k+1)\) — also pronic. They show up in many combinatorial counts including the number of edges in a complete bipartite graph \(K_{k, k+1}\).