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# Nonagonal Pentagonal Number

A number which is simultaneously a nonagonal number and pentagonal number and therefore satisfies the Diophantine equation

 (1)

Completing the square and rearranging gives

 (2)

Defining and gives the Pell-like equation

 (3)

This has solutions in corresponding to solutions which are integral in and of , (14, 21), (7189, 10981), (165026, 252081), (86968201, 132846121), ... (OEIS A048913 and A048914), giving the nonagonal pentagonal numbers 1, 651, 180868051, 95317119801, 26472137730696901, ... (OEIS A048915).

Nonagonal Number, Pentagonal Number

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## References

Sloane, N. J. A. Sequences A048913, A048914, and A048915 in "The On-Line Encyclopedia of Integer Sequences."

## Referenced on Wolfram|Alpha

Nonagonal Pentagonal Number

## Cite this as:

Weisstein, Eric W. "Nonagonal Pentagonal Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NonagonalPentagonalNumber.html