Cube as Sum of Sequence of Centered Hexagonal Numbers/Examples
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Examples of Cube as Sum of Sequence of Centered Hexagonal Numbers
\(\ds 1^3\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times 1 \paren {1 - 1} + 1\) |
\(\ds 2^3\) | \(=\) | \(\ds 8\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 + 7\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {3 \times 1 \paren {1 - 1} + 1}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {3 \times 2 \paren {2 - 1} + 1}\) |
\(\ds 3^3\) | \(=\) | \(\ds 27\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 + 7 + 19\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {3 \times 1 \paren {1 - 1} + 1}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {3 \times 2 \paren {2 - 1} + 1}\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {3 \times 3 \paren {3 - 1} + 1}\) |
\(\ds 4^3\) | \(=\) | \(\ds 64\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 + 7 + 19 + 37\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {3 \times 1 \paren {1 - 1} + 1}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {3 \times 2 \paren {2 - 1} + 1}\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {3 \times 3 \paren {3 - 1} + 1}\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {3 \times 4 \paren {4 - 1} + 1}\) |
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $64$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $64$