"On a Diophantine Equation" paper of Erdös, at some point it is said that it is well known that $C(n,2)=x^2$ has infinitely many integer solutions. I am just wondering the formula generating all possible $n \geq 2$,because I wonder whether they all come by integer solutions of $2x^2-y^2=\pm 1$ Pell's equation generated by fundamental unit, or is it possible to have different $n$'s that do not come from the solutions of the pell's equation. 


For example $C(2,2)=1$ a square, $C(9,2)=36$ a square too, so the first two $n$'s are 2,9.


One can also see from the solutions of $2x^2-y^2=\pm 1$ pell's equation, by take $x=1, y=0$ 
for $n=2=2\*1$ and take $x=2, y=3$ for $n=9=3^2$. 



Explicit solutions of C(n,2)=x^2 ?
Another approach to the problem, at least for non-number theorists, is to ask Mathematica:



> 
> Select[Range[10000], IntegerQ[Sqrt[Binomial[#, 2]]] &]
> 
> 
> 


and you find that the first examples are



> 
> {1, 2, 9, 50, 289, 1682, 9801}
> 
> 
> 


Then go to the OEIS (<https://oeis.org/>) and input that. You find that this is sequence A055997 (<https://oeis.org/A055997>), and the OEIS response is together with generating functions, recurrence relations, citations, and more.


One of those references is to an article titled "Discovering the Square-Triangular Numbers", which seems a promising title. The citation is to the Fibonacci Quarterly, and a little googling finds that their old issues are online (<http://www.fq.math.ca/list-of-issues.html>), and this particular article (by Phil Lafer) is, too (<http://www.fq.math.ca/Scanned/9-1/lafer.pdf>).


The article reads nicely (thank you Phil Lafer), and the Pell equation $2x^2-y^2=1$ does indeed make a fundamental appearance.


