On the impossibility of certain (n²+n+kₙ₊₁) configurations

This paper investigates the impossibility of certain (n²+n+kₙ₊₁) configurations. Firstly, for $k=2$, the result of {gropp1992non} that n²+n/2 is even and $n+1$ is a perfect square or n²+n/2 is odd and $n-1$ is a perfect square is reproved using the incidence matrix N and analysing the form of NTN. Then, for all k, configurations where paralellism is a transitive property are considered. It is then analogously established that if n≡0 or n≡ k-1 mod k for k even, then n²+n/k is even and $n+1$ is a perfect square or n²+n/k is odd and $n-(k-1)$ is a perfect square. Finally, the case $k=3$ is investigated in full generality.