A peaceful night for you,

I noticed for some quadratic polynomial, such as f(n)=n²+1, f(n)=2n²-1, f(n)=2n²+1 and f(n)=4n²+1 you can make a linear substitution with n=p*k+n0 with p|f(n) and p=f(n0) and a division by p so that you get

f(k)=ak²+bk+1

This seems to be something special. Do they have a mathematical name and which mathematician has investigated them ?

A short link would be really nice from you,

Greetings from Corona-times,

I live only in the night like a vampir