How do I prove the number of derangements formula $nD_{n - 1} + (-1)^n$ intuitively?

I found that the series of number of derangements follows a certain sequence that can be expressed as such : $$nD_{n-1} +(-1)^n$$.

Now, I found others saying it can be expressed as follows $$(n-1)(D_{n-1}+D_{n-2})$$.

I have seen some people explaining the second formula intuitively. What I am curious about is , whether there is a way to explain the FIRST formula intuitively. I want an explanation for the first formula. Meaning, I am trying to find the logic for it in such a way that the expression as it's written would be understood.

I started somewhat by saying that there is some logic behind the number of derangements of $$D_{n-1}$$ to be multiplied by $$n$$ because the new object has $$n$$ places to be placed in while $$n-1$$ objects are placed as was ordered in $$D_{n-1}$$. But I can't seem to find the logic behind the $$-1^n$$. I started to think that maybe for odd numbers I have to reduce one due to the middle? But I am not sure.

I want a logical/intuitive proof that would prove MY (first) formula and I want it to be followed somehow in the way I started. I don't want completely different proofs.

But can someone please make some sense of my proof or show why it's wrong? Or show other intuitive thing to my formula?