Heptagonal Triangular Number

A number which is simultaneously a heptagonal number  and triangular 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 basic solutions , (7, 3), and (18, 8). Additional solutions can be obtained from the unit Pell equation, and correspond to integer solutions when , (5, 10), (221, 493), (1513, 3382), ... (OEIS A046193 and A039835), corresponding to the heptagonal triangular numbers 1, 55, 121771, 5720653, 12625478965, ... (OEIS A046194).

REFERENCES:

Sloane, N. J. A. Sequences A039835, A046193, and A046194 in "The On-Line Encyclopedia of Integer Sequences."