Square pyramidal number

In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the number of stacked spheres in a pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part of a broader topic of figurate numbers representing the numbers of points forming regular patterns within different shapes. As well as counting spheres in a pyramid, these numbers can be described algebraically as a sum of the first positive square numbers, or as the values of a cubic polynomial. They can be used to solve several other counting problems, including counting squares in a square grid and counting acute triangles formed from the vertices of an odd regular polygon. They equal the sums of consecutive tetrahedral numbers, and are one-fourth of a larger tetrahedral number. The sum of two consecutive square pyramidal numbers is an octahedral number. The pyramidal numbers were one of the few types of three-dimensional figurate numbers studied in Greek mathematics, in works by Nicomachus, Theon of Smyrna, and Iamblichus. Formulas for summing consecutive squares to give a cubic polynomial, whose values are the square pyramidal numbers, are given by Archimedes, who used this sum as a lemma as part of a study of the volume of a cone, and by Fibonacci, as part of a more general solution to the problem of finding formulas for sums of progressions of squares. The square pyramidal numbers were also one of the families of figurate numbers studied by Japanese mathematicians of the wasan period, who named them "kirei saijō suida" (with modern kanji, 奇零 再乗 蓑深). The same problem, formulated as one of counting the cannonballs in a square pyramid, was posed by Walter Raleigh to mathematician Thomas Harriot in the late 1500s, while both were on a sea voyage. The cannonball problem, asking whether there are any square pyramidal numbers that are also square numbers other than 1 and 4900, is said to have developed out of this exchange.