Equivalence of Definitions of Square Number

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

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

and so by Definition 2:
 * $\ds \sum_{j \mathop = 1}^n \paren {2 j - 1} = S_{n-1} + 2 n - 1$

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

Then:

Thus $\map P {4, n}$ and $S_n$ are generated by the same recurrence relation.