As answered by nadeemoidu

 

3n²=k(k+1)

3=k(k+1)/n²   ...........................................1 

As k & k+1 have no common factors, either k or k+1 is divisible by n²

let k=xn² for some integer x

so (1) comes to 3=x(xn²+1)

So either x=3 & xn²+1=1 OR x=1 & xn²+1=3 [as 3 is prime]

in both the cases n is not an integer

 

Hence k+1=xn²

substituting k+1 in (1)  3=x(xn²-1)

for x=1 and xn²-1=3   n² =4, which is the only soln.

 

So 2+2sqrt(1+12n²)=16=4²