Square Pyramidal Number - Formula

Formula

The first few square pyramidal numbers are:

1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819 (sequence A000330 in OEIS).

These numbers can be expressed in a formula as

This is a special case of Faulhaber's formula, and may be proved by a straightforward 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)(2t + 3)/6 = Pt + 1.

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