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.
Read more about this topic: Square Pyramidal Number
Famous quotes containing the word formula:
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—Friedrich Nietzsche (1844–1900)
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“Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective positions of the beings which compose it, if moreover this intelligence were vast enough to submit these data to analysis, it would embrace in the same formula both the movements of the largest bodies in the universe and those of the lightest atom; to it nothing would be uncertain, and the future as the past would be present to its eyes.”
—Pierre Simon De Laplace (1749–1827)