Nonagonal Hexagonal Number

A number which is simultaneously a nonagonal number  and hexagonal 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 fundamental solutions , (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  and  of , (10, 13), (39025, 51625), ... (OEIS A048916 and A048917), giving the nonagonal hexagonal numbers 1, 325, 5330229625, 1353857339341, 22184715227362706161, ... (OEIS A048918).

REFERENCES:

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