Octagonal Heptagonal Number

A number which is simultaneously octagonal and heptagonal. Let  denote the th octagonal number and  the th heptagonal number, then a number which is both octagonal and hexagonal satisfies the equation , or

(1)

Completing the square and rearranging gives

(2)

Therefore, defining

			(3)
			(4)

gives the second-order Diophantine equation

(5)

The first few solutions are , (7, 4), (73, 40), (157, 86), .... These give the integer solutions (1, 1), (345, 315), (166145, 151669), ... (OEIS A048904 and A048905), corresponding to the octagonal heptagonal numbers 1, 297045, 69010153345, ... (OEIS A048906).

REFERENCES:

Sloane, N. J. A. Sequences A048904, A048905, and A048906 in "The On-Line Encyclopedia of Integer Sequences."