How many integral solutions (x, y) exist satisfying the equation |y| + |x| ≤ 4 

My approach:

I have made the graph after opening the the modulus in the above equation by making four equations.

Now it is a square with co-ordinates (4,0)(0,4)(-4,0)(0,-4).

Now I am stuck and don't know how to calculate integral solutions. It should be integral boundary points plus the integral points inside the area.

I know about the Pick's theorem in which we can find the integral points by using area and boundary points but I need to know how to calculate the integral points without it.

Answer is 41.
In my book it is given as 9+2(7+5+3+1)=41 [which I am not getting]

Kindly help in solving the same.