# Deriving difference equation

I have this block-diagram of Karplus-Strong basic algorithm.

Where $$H_{d}(z) = \frac{1+z^{-1}}{2}$$ The problem occurs when I try to get the difference equation. As I can see from the diagram, that would be $$y(n) = x(n-N)+\frac{y(n)+y(n-1)}{2}$$ but I've found online this one(which I know how to implement in code, unlike the first one)$$y(n) = x(n)+\frac{y(n-N)+y(n-(N+1))}{2}$$ Are these two equation somehow equivalent?

$$x[n]+\frac12\big(y[n]+y[n-1]\big)\tag{1}$$
Then a delay of $$N$$ samples is applied to compute the output, hence
$$y[n]=x[n-N]+\frac12\big(y[n-N]+y[n-1-N]\big)\tag{2}$$