Hexagonal Pentagonal Number

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

(1)

Completing the square and rearranging gives

(2)

Therefore, defining

			(3)
			(4)

gives the Pell-like equation

(5)

The first few solutions are , (5, 3), (19, 11), (71, 41), (265, 153), (989, 571), .... These give the solutions , (1, 1), (10/3, 3), (12, 21/2), (133/3, 77/2), (165, 143), ..., of which the integer solutions are (1, 1), (165, 143), (31977, 27693), (6203341, 5372251), ... (OEIS A046178 and A046179), corresponding to the pentagonal hexagonal numbers 1, 40755, 1533776805, 57722156241751, ... (OEIS A046180).

REFERENCES:

Sloane, N. J. A. Sequences A046178, A046179, and A046180 in "The On-Line Encyclopedia of Integer Sequences."