Nonagonal Heptagonal Number

A number which is simultaneously a nonagonal number  and heptagonal 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 , (88, 104), (12445, 14725), (1767052, 2090804), ... (OEIS A048919 and A048920), giving the nonagonal heptagonal numbers 1, 26884, 542041975, 10928650279834, ... (OEIS A048921).

REFERENCES:

Sloane, N. J. A. Sequences A048919, A048920, and A048921 in "The On-Line Encyclopedia of Integer Sequences."