Square pyramidal number

In mathematics, a pyramid number, or square pyramidal number, is a figurate number that represents the number of stacked spheres in a pyramid with a square base. Square pyramidal numbers also solve the problem of counting the number of squares in an $n \times n$ grid.

The first few square pyramidal numbers are:

These numbers can be expressed in a formula as

This is a special case of Faulhaber's formula, and may be proved by a mathematical induction. An equivalent formula is given in Fibonacci's Liber Abaci (1202, ch. II.12).

In modern mathematics, figurate numbers are formalized by the Ehrhart polynomials. The Ehrhart polynomial $L (P, t)$ of a polyhedron $P$ is a polynomial that counts the number of integer points in a copy of $P$ that is expanded by multiplying all its coordinates by the number $t$ . The Ehrhart polynomial of a pyramid whose base is a unit square with integer coordinates, and whose apex is an integer point at height one above the base plane, is $(t + 1)(t + 2)(2 t + 3)/6 = P t + 1$ .

The square pyramidal numbers can also be expressed as sums of binomial coefficients:

The binomial coefficients occurring in this representation are tetrahedral numbers, and this formula expresses a square pyramidal number as the sum of two tetrahedral numbers in the same way as square numbers are the sums of two consecutive triangular numbers.

The smaller tetrahedral number represents $1+3+6+\dots +T(n+1)$ $1+3+6+\dots +T(n+1)$ and the larger $1+3+6+\dots +T(n+2)$ $1+3+6+\dots +T(n+2)$ . Offsetting the larger and adding, we arrive at $1,1+3,3+6\dots$ $1,1+3,3+6\dots$ , the square numbers.

...
Wikipedia