Cube as Sum of Sequence of Centered Hexagonal Numbers
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Theorem
- $C_n = \ds \sum_{i \mathop = 1}^n H_i$
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:
\(\ds \sum_{i \mathop = 1}^n H_i\) | \(=\) | \(\ds \sum_{i \mathop = 1}^n \paren {3 i \paren {i - 1} + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^n \paren {3 i^2 - 3 i + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \sum_{i \mathop = 1}^n i^2 - 3 \sum_{i \mathop = 1}^n i + \sum_{i \mathop = 1}^n 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \frac {n \paren {n + 1} \paren {2 n + 1} } 6 - 3 \sum_{i \mathop = 1}^n i + \sum_{i \mathop = 1}^n 1\) | Sum of Sequence of Squares | |||||||||||
\(\ds \) | \(=\) | \(\ds 3 \frac {n \paren {n + 1} \paren {2 n + 1} } 6 - 3 \frac {n \paren {n + 1} } 2 + \sum_{i \mathop = 1}^n 1\) | Closed Form for Triangular Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {n \paren {\paren {n + 1} \paren {2 n + 1} - 3 \paren {n + 1} + 2} } 2\) | simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {n \paren {2 n^2 + 3 n + 1 - 3 n - 3 + 2} } 2\) | multiplying out | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {n \paren {2 n^2} } 2\) | simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds n^3\) | simplification |
$\blacksquare$
Examples
\(\ds 1^3\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times 1 \paren {1 - 1} + 1\) |
\(\ds 2^3\) | \(=\) | \(\ds 8\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 + 7\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {3 \times 1 \paren {1 - 1} + 1}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {3 \times 2 \paren {2 - 1} + 1}\) |
\(\ds 3^3\) | \(=\) | \(\ds 27\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 + 7 + 19\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {3 \times 1 \paren {1 - 1} + 1}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {3 \times 2 \paren {2 - 1} + 1}\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {3 \times 3 \paren {3 - 1} + 1}\) |
\(\ds 4^3\) | \(=\) | \(\ds 64\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 + 7 + 19 + 37\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {3 \times 1 \paren {1 - 1} + 1}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {3 \times 2 \paren {2 - 1} + 1}\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {3 \times 3 \paren {3 - 1} + 1}\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {3 \times 4 \paren {4 - 1} + 1}\) |
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $64$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $64$