Nonagonal Octagonal Number

A number which is simultaneously a nonagonal number  and octagonal number  and therefore satisfies the Diophantine equation

(1)

Completing the square and rearranging gives

(2)

Defining  and  gives the Pell-like equation

(3)

The first integral solutions in  and  are , (425, 459), (286209, 309141), (192904201, 208360351), ... (OEIS A048922 and A048923), giving the nonagonal octagonal numbers 1, 631125, 286703855361, 130242107189808901, ... (OEIS A048924).

REFERENCES:

Sloane, N. J. A. Sequences A048922, A048923, and A048924 in "The On-Line Encyclopedia of Integer Sequences."