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), ... (Sloane's A046181 and A046182), corresponding to the octagonal triangular numbers 1, 21, 11781, 203841, 113123361, ... (Sloane's A046183).

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

CITE THIS AS:

Weisstein, Eric W. "Octagonal Triangular Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/OctagonalTriangularNumber.html