Find a polynomial as $2836x^2-12724x+16129$ I found a polynomial function with integer coefficients:$f(x)=2836x^2-12724x+16129$
and $f(0)=127^2,f(1)=79^2,f(2)=45^2,f(3)=59^2,f(4)=103^2,f(5)=153^2.$
My question is:can we find a polynomial function with integer coefficients,called $f(x)$,which has no multiple roots,and $f(0),f(1),f(2),f(3),……,f(k)$ are distinct square numbers?($k>5$ is a given integer)
Thanks all.
PS:I'm sorry,guys.I lost a very important condition:$f(x)$ should be a quadratic function:$f(x)=ax^2+bx+c$.($a,b,c$ are integers and $b^2-4ac≠0$)
So the Lagrange interpolation method does not work.
I wonder is there always such a quadratic polynomial when $k$ is arbitrarily large? 

One such quadratic 
$$p(t)=-4980t^2+32100t+2809$$
$p(0)=53^2,p(1)=173^2,p(2)=217^2,p(3)=233^2,p(4)=227^2,p(5)=197^2,p(6)=127^2$
Source : Polynomials E.J Barbeau
