Figurate Number Calculator

T(n) = n(n+1)(n+2)/6 = C(n+2,3). They count balls in a tetrahedral pile: 1 on top, 3 in next layer, 6 in next, etc. T(n) = sum of first n triangular numbers. Named because the balls form a tetrahedron.

Is there a square pyramidal number that's also a perfect square? Yes: 4900 = P(24) = 70². Proved by Watson (1918) that this is the only non-trivial solution. Also called the cannonball problem since cannonballs stacked in a square pyramid somehow fill a square.

Regular polygonal numbers count dots in a corner-based arrangement. Centered polygonal numbers count dots in concentric rings around a center dot. Centered hexagonal(n) = 3n²−3n+1 vs hexagonal(n) = n(2n−1). They model different geometric patterns.