Equivalence of Definitions of Square Number

Theorem
The following definitions of a square number are equivalent:

Definition 1 equivalent to Definition 3
By the Odd Number Theorem:
 * $\displaystyle \sum_{j \mathop = 1}^n \left({2 j - 1}\right) = n^2$

Definition 2 equivalent to Definition 3
By the Corollary to the Odd Number Theorem:
 * $S_n = \displaystyle \sum_{j \mathop = 1}^{n - 1} + 2 n - 1$

and so by Definition 2:
 * $\displaystyle \sum_{j \mathop = 1}^n \left({2 j - 1}\right) = S_{n-1} + 2 n - 1$

Definition 2 equivalent to Definition 4
We have by definition that $S_n = 0 = P \left({4, n}\right)$.

Then:

Thus $P \left({4, n}\right)$ and $S_n$ are generated by the same recurrence relation.