Notes on polynomials $(1+X{)}^n+(-1{)}^n(X^n+1)$ concerning the regularity problem for symmetric power sums in 3 variables

Let $K$ be a field and $f _{n}(X) = (X + 1) ^{n} + (-1) ^{n}(X ^{n} + 1) \in K[X]$,foreach$n \in \mathbb N$\mathrm{.}Thisnoteshowsthatthepolynomials$f _{m}(X)$and$f _{m’}(X)$arerelativelyprime,forsomedistinctindices$m$and$m ^{\prime}$atmostequalto$100$,ifandonlyiftheproduct$mm ^{\prime }$isdivisibleby$6$.