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Nonagonal Hexagonal Number


A number which is simultaneously a nonagonal number N_m and hexagonal number Hex_n and therefore satisfies the Diophantine equation

 1/2m(7m-5)=n(2n-1).
(1)

Completing the square and rearranging gives

 (14n-5)^2-7(4m-1)^2=18.
(2)

Defining x=14n-5 and y=4m-1 gives the Pell-like equation

 x^2-7y^2=18.
(3)

This has fundamental solutions (x,y)=(5,1), (9, 3), and (19, 17), giving the family of solutions (5, 1), (9, 3), (19, 17), (61, 23), (135, 51), (509, 193), .... These give solutions which are integers in m and n of (m,n)=(1,1), (10, 13), (39025, 51625), ... (OEIS A048916 and A048917), giving the nonagonal hexagonal numbers 1, 325, 5330229625, 1353857339341, 22184715227362706161, ... (OEIS A048918).


See also

Hexagonal Number, Nonagonal Number

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References

Sloane, N. J. A. Sequences A048916, A048917, and A048918 in "The On-Line Encyclopedia of Integer Sequences."

SeeAlso

Nonagonal Number

Referenced on Wolfram|Alpha

Nonagonal Hexagonal Number

Cite this as:

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

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