Cube as Sum of Sequence of Centered Hexagonal Numbers

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Theorem

$C_n = \displaystyle \sum_{i \mathop = 1}^n H_i$

where:

$C_n$ denotes the $n$th cube number
$H_i$ denotes the $i$th centered hexagonal number.


Proof

From Closed Form for Centered Hexagonal Numbers:

$H_n = 3 n \paren {n - 1} + 1$

Hence:

\(\displaystyle \sum_{i \mathop = 1}^n H_i\) \(=\) \(\displaystyle \sum_{i \mathop = 1}^n \paren {3 i \paren {i - 1} + 1}\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 1}^n \paren {3 i^2 - 3 i + 1}\)
\(\displaystyle \) \(=\) \(\displaystyle 3 \sum_{i \mathop = 1}^n i^2 - 3 \sum_{i \mathop = 1}^n i + \sum_{i \mathop = 1}^n 1\)
\(\displaystyle \) \(=\) \(\displaystyle 3 \frac {n \paren {n + 1} \paren {2 n + 1} } 6 - 3 \sum_{i \mathop = 1}^n i + \sum_{i \mathop = 1}^n 1\) Sum of Sequence of Squares
\(\displaystyle \) \(=\) \(\displaystyle 3 \frac {n \paren {n + 1} \paren {2 n + 1} } 6 - 3 \frac {n \paren {n + 1} } 2 + \sum_{i \mathop = 1}^n 1\) Closed Form for Triangular Numbers
\(\displaystyle \) \(=\) \(\displaystyle \frac {n \paren {\paren {n + 1} \paren {2 n + 1} - 3 \paren {n + 1} + 2} } 2\) simplification
\(\displaystyle \) \(=\) \(\displaystyle \frac {n \paren {2 n^2 + 3 n + 1 - 3 n - 3 + 2} } 2\) multiplying out
\(\displaystyle \) \(=\) \(\displaystyle \frac {n \paren {2 n^2} } 2\) simplification
\(\displaystyle \) \(=\) \(\displaystyle n^3\) simplification

$\blacksquare$


Examples

\(\displaystyle 1^3\) \(=\) \(\displaystyle 1\)
\(\displaystyle \) \(=\) \(\displaystyle 3 \times 1 \paren {1 - 1} + 1\)


\(\displaystyle 2^3\) \(=\) \(\displaystyle 8\)
\(\displaystyle \) \(=\) \(\displaystyle 1 + 7\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {3 \times 1 \paren {1 - 1} + 1}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \paren {3 \times 2 \paren {2 - 1} + 1}\)


\(\displaystyle 3^3\) \(=\) \(\displaystyle 27\)
\(\displaystyle \) \(=\) \(\displaystyle 1 + 7 + 19\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {3 \times 1 \paren {1 - 1} + 1}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \paren {3 \times 2 \paren {2 - 1} + 1}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \paren {3 \times 3 \paren {3 - 1} + 1}\)


\(\displaystyle 4^3\) \(=\) \(\displaystyle 64\)
\(\displaystyle \) \(=\) \(\displaystyle 1 + 7 + 19 + 37\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {3 \times 1 \paren {1 - 1} + 1}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \paren {3 \times 2 \paren {2 - 1} + 1}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \paren {3 \times 3 \paren {3 - 1} + 1}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \paren {3 \times 4 \paren {4 - 1} + 1}\)


Sources