Cube as Sum of Sequence of Centered Hexagonal Numbers/Examples

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): $64$