Nonagonal Hexagonal Number

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


Completing the square and rearranging gives


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


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

Explore with Wolfram|Alpha


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


Nonagonal Number

Referenced on Wolfram|Alpha

Nonagonal Hexagonal Number

Cite this as:

Weisstein, Eric W. "Nonagonal Hexagonal Number." From MathWorld--A Wolfram Web Resource.

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