OEIS A000330: Square pyramidal numbers

In this episode we dive into A000330, the square pyramidal numbers, defined by a(n) = 0^2 + 1^2 + 2^2 + ... + n^2 = n*(n+1)*(2*n+1)/6. We’ll see why these count cannonball pyramids with square bases and, in the 2D analogue, the total number of squares in an n×n grid. We discuss the key identity S(n) = T(n) + T(n−1), where T(k) are tetrahedral numbers, linking square pyramidal numbers to other figurate families. We’ll cover famous results: the only square pyramidal number greater than 1 that is also a perfect square is 4900, and no square pyramidal number greater than 1 is tetrahedral. We’ll also explore interesting number-theoretic properties—units digits form a period-20 cycle, and n divides S(n) iff n ≡ ±1 (mod 6). Finally, we glimpse a tantalizing conjecture that every integer can be expressed as a sum of three generalized square pyramidal numbers. A rich tour of geometry, combinatorics, and modular arithmetic awaits.

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