Octagonal Pentagonal Number

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

(1)

Completing the square and rearranging gives

(2)

Therefore, defining

			(3)
			(4)

gives the Pell equation

(5)

The first few solutions are , (5, 4), (11, 8), (31, 22), (65, 46), .... These give the solutions , (1, 1), (2, 5/3), (16/3, 4), (11, 8), ..., of which the integer solutions are (1, 1), (11, 8), (1025, 725), (12507, 8844), ... (OEIS A046187 and A046188), corresponding to the octagonal pentagonal numbers 1, 176, 1575425, 234631320, 2098015778145, ... (OEIS A046189).

REFERENCES:

Sloane, N. J. A. Sequences A046187, A046188, and A046188 in "The On-Line Encyclopedia of Integer Sequences."