# Sum of Sequence of Odd Squares/Formulation 1

$\ds \forall n \in \N: \sum_{i \mathop = 0}^n \paren {2 i + 1}^2 = \frac {\paren {n + 1} \paren {2 n + 1} \paren {2 n + 3} } 3$
 $\ds \sum_{i \mathop = 0}^n \paren {2 i + 1}^2$ $=$ $\ds \sum_{i \mathop = 0}^n \paren {2 i}^2 + \sum_{i \mathop = 0}^n 4 i + \sum_{i \mathop = 0}^n 1$ $\ds$ $=$ $\ds \frac {2 n \paren {n + 1} \paren {2 n + 1} } 3 + 4 \sum_{i \mathop = 0}^n i + \sum_{i \mathop = 0}^n 1$ Sum of Sequence of Even Squares $\ds$ $=$ $\ds \frac {2 n \paren {n + 1} \paren {2 n + 1} } 3 + 4 \sum_{i \mathop = 1}^n i + \sum_{i \mathop = 0}^n 1$ adjustment of indices: $4 i = 0$ when $i = 0$ $\ds$ $=$ $\ds \frac {2 n \paren {n + 1} \paren {2 n + 1} } 3 + 4 \frac {n \paren {n + 1} } 2 + \sum_{i \mathop = 0}^n 1$ Closed Form for Triangular Numbers $\ds$ $=$ $\ds \frac {2 n \paren {n + 1} \paren {2 n + 1} } 3 + 2 n \paren {n + 1} + \paren {n + 1}$ further simplification $\ds$ $=$ $\ds \frac {\paren {n + 1} \paren {2 n \paren {2 n + 1} + 6 n + 3} } 3$ factorising $\ds$ $=$ $\ds \frac {\paren {n + 1} \paren {4 n^2 + 8 n + 3} } 3$ multiplying out $\ds$ $=$ $\ds \frac {\paren {n + 1} \paren {2 n + 1} \paren {2 n + 3} } 3$ factorising
$\blacksquare$