Sequence of Square Centered Hexagonal Numbers

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Theorem

The sequence of centered hexagonal numbers which are also square begins:

$1, 169, 32 \, 761, 6 \, 355 \, 441, \ldots$

This sequence is A006051 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

The sequence of centered hexagonal numbers, for $n \in \Z_{> 0}$, begins:

$1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, \ldots$


We have that:

\(\ds 1\) \(=\) \(\ds 1^2\) Definition of Square Number
\(\ds \) \(=\) \(\ds 3 \times 1 \times \paren {1 - 1} + 1\) Closed Form for Centered Hexagonal Numbers


\(\ds 169\) \(=\) \(\ds 13^2\) Definition of Square Number
\(\ds \) \(=\) \(\ds 3 \times 8 \times \paren {8 - 1} + 1\) Closed Form for Centered Hexagonal Numbers


\(\ds 32 \, 761\) \(=\) \(\ds 181^2\) Definition of Square Number
\(\ds \) \(=\) \(\ds 3 \times 105 \times \paren {105 - 1} + 1\) Closed Form for Centered Hexagonal Numbers


\(\ds 6 \, 355 \, 441\) \(=\) \(\ds 2521^2\) Definition of Square Number
\(\ds \) \(=\) \(\ds 3 \times 1456 \times \paren {1456 - 1} + 1\) Closed Form for Centered Hexagonal Numbers

$\blacksquare$


Sources

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