Pentagonal Triangular Number

A number which is simultaneously a pentagonal number  and triangular number . Such numbers exist when

(1)

Completing the square gives

(2)

Substituting  and  gives the Pell-like quadratic Diophantine equation

(3)

which has solutions , (19, 11), (71, 41), (265, 153), .... In terms of , these give (1, 1), (10/3,5), (12, 20), (133/3, 76), (165, 285), ..., of which the whole number solutions are , (12, 20), (165, 285), (2296, 3976), ... (OEIS A046174 and A046175), corresponding to the pentagonal triangular numbers 1, 210, 40755, 7906276, 1533776805, ... (OEIS A014979).

REFERENCES:

Silverman, J. H. A Friendly Introduction to Number Theory. Englewood Cliffs, NJ: Prentice Hall, 1996.

Sloane, N. J. A. Sequences A014979, A046174, and A046175 in "The On-Line Encyclopedia of Integer Sequences."