Heptagonal Pentagonal Number

A number which is simultaneously a heptagonal number  and pentagonal 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, 5), (17, 13), (53, 41), (133, 103), .... The integer solutions in  and  are then given by , (42, 54), (2585, 3337), (160210, 206830), (9930417, 12820113) ... (OEIS A046198 and A046199), corresponding to the heptagonal pentagonal numbers 1, 4347, 16701685, 64167869935, 246532939589097, ... (OEIS A048900).

REFERENCES:

Sloane, N. J. A. Sequences A046198, A046199, and A048900 in "The On-Line Encyclopedia of Integer Sequences."