Octagonal Triangular Number

A number which is simultaneously octagonal and triangular. Let  denote the th octagonal number and  the th triangular number, then a number which is both octagonal and triangular 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, 6), (20/3, 15), (80/3, 64), (63, 153), ..., of which the integer solutions are (1, 1), (3, 6), (63, 153), (261, 638), (6141, 15041), (25543, 62566), (601723, 1473913), ... (OEIS A046181 and A046182), corresponding to the octagonal triangular numbers 1, 21, 11781, 203841, 113123361, ... (OEIS A046183).

REFERENCES:

Sloane, N. J. A. Sequences A046181, A046182, and A046183 in "The On-Line Encyclopedia of Integer Sequences."