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:
“The formula for achieving a successful relationship is simple: you should treat all disasters as if they were trivialities but never treat a triviality as if it were a disaster.”
—Quentin Crisp (b. 1908)
““It’s hard enough to adjust [to the lack of control] in the beginning,” says a corporate vice president and single mother. “But then you realize that everything keeps changing, so you never regain control. I was just learning to take care of the belly-button stump, when it fell off. I had just learned to make formula really efficiently, when Sarah stopped using it.””
—Anne C. Weisberg (20th century)
“I take it that what all men are really after is some form or perhaps only some formula of peace.”
—Joseph Conrad (1857–1924)