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There are many integer points on the hyperboloid of two sheets $x^2+y^2-z^2=-1$.

(0,0,1), (2,2,3), (4,8,9),...

Let us denote such set as H.

I will consider only the upper sheet $z>0$, but sign variations in x and y are in it. The number of such points with z coordinate below or at a certain value n will be denoted by P(n)

$P(n) = |(x,y,z) \in H, 0<z\leq n|$

as an example

P(1)=1 ... because only (0,0,1) fits the description
P(2)=1 ... we don't get anything else
P(3)=5 ... we pick up 4 sign variations of (2,2,3)
...

I was wondering how quickly this sequence grows and it seems that P(n) is very close to n, asymptotically speaking. Is there a reason for it? What are good bounds for $|P(n)-n|$?

For the first half milion values I calculated it below.

empirical data

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