Square Pyramidal Number - Relations To Other Figurate Numbers

Relations To Other Figurate Numbers

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. In this sum, one of the two tetrahedral numbers counts the number of balls in a stacked pyramid that are directly above or to one side of a diagonal of the base square, and the other tetrahedral number in the sum counts the number of balls that are to the other side of the diagonal. Square pyramidal numbers are also related to tetrahedral numbers in a different way:

The sum of two consecutive square pyramidal numbers is an octahedral number.

Augmenting a pyramid whose base edge has n balls by adding to one of its triangular faces a tetrahedron whose base edge has n − 1 balls produces a triangular prism. Equivalently, a pyramid can be expressed as the result of subtracting a tetrahedron from a prism. This geometric dissection leads to another relation:

Besides 1, there is only one other number that is both a square and a pyramid number: 4900, which is both the 70th square number and the 24th square pyramidal number. This fact was proven by G. N. Watson in 1918.

Another relationship involves the Pascal Triangle: Whereas the classical Pascal Triangle with sides (1,1) has diagonals with the natural numbers, triangular numbers, and tetrahedral numbers, generating the Fibonacci numbers as sums of samplings across diagonals, the sister Pascal with sides (2,1) has equivalent diagonals with odd numbers, square numbers, and square pyramidal numbers, respectively, and generates (by the same procedure) the Lucas numbers rather than Fibonacci.

In the same way that the square pyramidal numbers can be defined as a sum of consecutive squares, the squared triangular numbers can be defined as a sum of consecutive cubes.