Hexagonal Square Number

Let  denote the th hexagonal number and  the th square number, then a number which is both hexagonal and square 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 , (17, 12), (99, 70), (577, 408), .... These give the solutions , (9/2, 6), (25, 35), (289/2, 204), ..., giving the integer solutions (1, 1), (25, 35), (841, 1189), (28561, 40391), ... (OEIS A008844 and A046176). The corresponding hexagonal square numbers are 1, 1225, 1413721, 1631432881, 1882672131025, ... (OEIS A046177).

Closed-form solutions are

			(6)
			(7)

giving the th hexagonal square number as

(8)

A recurrence relation for  is given by

(9)

with , where  (M. Carreira, pers. comm., Sept. 11, 2004).

REFERENCES:

Sloane, N. J. A. Sequences A008844, A046176, and A046177 in "The On-Line Encyclopedia of Integer Sequences."