A number which is simultaneously a nonagonal number and hexagonal number and therefore satisfies the Diophantine equation
(1)
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Completing the square and rearranging gives
(2)
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Defining and gives the Pell-like equation
(3)
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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).