> Let $n$ be a positive integer. Determine the smallest possible value of $|p(1)|^2+|p(2)|^2 +...+ |p(n+3)|^2$ over all monic polynomials $p$ of degree $n$.

This question was proposed (problem A.611)
some time ago at [KoMaL](http://www.komal.hu/verseny/feladat.cgi?a=feladat&f=A611&l=en).  
The minimal values for $n=0,1,2,3$ are $3,5,14,324/5$.  
It was also discussed at [math.se](http://math.stackexchange.com/questions/796808/minimising-an-expression-involving-polynomial), where the polynomial minimising such sum was found using Gram-Schmidt algorithm.  
However the minimum value was not determined.

Any suggestion is welcome.