Octagonal Hexagonal Number

A number which is simultaneously octagonal and hexagonal. Let  denote the th octagonal number and  the th hexagonal 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 , (4, 3), (16, 13), (38, 31), (158, 129), (376, 307), .... These give the solutions , (1, 1), (3, 7/2), (20/3, 8), (80/3, 65/2), (63, 77), ..., of which the integer solutions are (1, 1), (63, 77), (6141, 7521), (601723, 736957), ... (OEIS A046190 and A046191), corresponding to the octagonal hexagonal numbers 1, 11781, 113123361, 1086210502741, ... (OEIS A046192).

REFERENCES:

Sloane, N. J. A. Sequences A046190, A046191, and A046192 in "The On-Line Encyclopedia of Integer Sequences."