Triangular Numbers which are also Pentagonal
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Theorem
The sequence of triangular numbers which are also pentagonal begins:
- $1, 210, 40 \, 755, 7 \, 906 \, 276, 1 \, 533 \, 776 \, 805, 297 \, 544 \, 793 \, 910, \ldots$
This sequence is A014979 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
\(\ds 1\) | \(=\) | \(\ds \dfrac {1 \paren {3 \times 1 - 1} } 2\) | Closed Form for Pentagonal Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {1 \times \paren {1 + 1} } 2\) | Closed Form for Triangular Numbers |
\(\ds 210\) | \(=\) | \(\ds \dfrac {12 \paren {3 \times 12 - 1} } 2\) | Closed Form for Pentagonal Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {20 \times \paren {20 + 1} } 2\) | Closed Form for Triangular Numbers |
\(\ds 40 \, 755\) | \(=\) | \(\ds \dfrac {165 \paren {3 \times 165 - 1} } 2\) | Closed Form for Pentagonal Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {285 \times \paren {285 + 1} } 2\) | Closed Form for Triangular Numbers |
This theorem requires a proof. In particular: It remains to be shown that these are the only such instances. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $210$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $210$