Nonagonal Triangular Number

A number which is simultaneously a nonagonal number  and a triangular 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 unit solutions , (9, 3), and (19, 7), which lead to the family of solutions (5, 1), (9, 3), (19, 7), (61, 23), (135, 51), (299, 113), (971, 367), .... The corresponding integer solutions in  and  are , (10, 25), (154, 406), (2449, 6478), ... (OEIS A048907 and A048908), giving the nonagonal triangular numbers 1, 325, 82621, 20985481, 5330229625, 1353857339341, ... (OEIS A048909).

REFERENCES:

Sloane, N. J. A. Sequences A048907, A048908, and A048909 in "The On-Line Encyclopedia of Integer Sequences."