Nicomachus's Theorem

Nicomachus noted that when the odd numbers are grouped in blocks of length, 1, 2, 3, ..., i.e., 1; 3+5; 7+9+11; 13+15+17+19; ..., the th cubic number is a sum of consecutive odd numbers, for example

			(1)
			(2)
			(3)
			(4)

and so on (Merzbach and Boyer 1991, p. 160).

This set of identities that follows from

(5)

When coupled with the early Pythagorean recognition that the sum of the first odd numbers is (Merzbach and Boyer 1991, p. 160), this leads to the case of Faulhaber's formula

			(6)
			(7)
			(8)

that is sometimes known as Nicomachus's theorem. For , 2, ..., the first few values are 1, 9, 36, 100, 225, 441, 784, 1296, 2025, ... (OEIS A000537), which are squared triangular numbers (although not square triangular numbers).

Plugging in gives 2025, a value that which will not correspond to the current calendar year for another millennium.