Heptagonal Hexagonal Number

A number which is simultaneously a heptagonal number  and hexagonal number . Such numbers exist when

(1)

Completing the square and rearranging gives

(2)

Substituting  and  gives the Pell-like quadratic Diophantine equation

(3)

which has solutions , (7, 3), (18, 8), (47, 21), (123, 55), .... The integer solutions in  and  are then given by , (221, 247), (71065, 79453), (22882613, 25583539), ... (OEIS A048902 and A048901), corresponding to the heptagonal hexagonal numbers 1, 121771, 12625478965, 1309034909945503, ... (OEIS A048903).

REFERENCES:

Sloane, N. J. A. Sequences A048901, A048902, and A048903 in "The On-Line Encyclopedia of Integer Sequences."