I want to prove that this language is not regular $L = a^nb^{n^2+n}; n \geq 0$
It proved a bit challenging to prove it directly, and thus I am looking for the intersection way.
I want a regular $L_1$ such that $L \cap L_1 = L_2$ such that $L_2$ is not regular as well, $L_2$ should be something easy to prove and we know. This way it might be simpler to prove $L$ is not regular. Could you help with this please?
For example I proved $L_3 = a^nb^{n^2}$ not regular the following way.
We have $L_3 \cap b^* = b^{n^2}$, we know that $b^*$ is regular, thus, if $L_3$ is regular, then $b^{n^2}$ is regular, but since $b^{n^2}$ is not, then, $L_3$ is not.
But I don't know how to approach $L$ in a similar way, because directly it is not easy.