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Octagonal Heptagonal Number


A number which is simultaneously octagonal and heptagonal. Let O_m denote the mth octagonal number and H_n the nth heptagonal number, then a number which is both octagonal and hexagonal satisfies the equation H_n=O_m, or

 1/2n(5n-3)=m(3m-2).
(1)

Completing the square and rearranging gives

 3(10n-3)^2-40(3m-1)^2=-13.
(2)

Therefore, defining

x=(10n-3)
(3)
y=2(3m-1)
(4)

gives the second-order Diophantine equation

 3x^2-10y^2=-13
(5)

The first few solutions are (x,y)=(3,2), (7, 4), (73, 40), (157, 86), .... These give the integer solutions (1, 1), (345, 315), (166145, 151669), ... (OEIS A048904 and A048905), corresponding to the octagonal heptagonal numbers 1, 297045, 69010153345, ... (OEIS A048906).


See also

Heptagonal Number, Octagonal Number

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References

Sloane, N. J. A. Sequences A048904, A048905, and A048906 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Octagonal Heptagonal Number

Cite this as:

Weisstein, Eric W. "Octagonal Heptagonal Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OctagonalHeptagonalNumber.html

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