Pronic Number

A Figurate Number of the form $P_n=2T_n=n(n+1)$, where $T_n$ is the $n$th Triangular Number. The first few are 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, ... (Sloane's A002378). The Generating Function of the pronic numbers is

$${2x\over(1-x)^3}=2x+6x^2+12x^3+20x^4+\ldots.$$

The first few $n$ for which $P_n$ are Palindromic are 1, 2, 16, 77, 538, 1621, ... (Sloane's A028336), and the first few Palindromic Numbers which are pronic are 2, 6, 272, 6006, 289982, ... (Sloane's A028337).

References

De Geest, P. ``Palindromic Products of Two Consecutive Integers.'' http://www.ping.be/~ping6758/consec.htm.

Sloane, N. J. A. A028336, A028337, and A002378/M1581 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.