Haskell, 41 bytes
f n=1+4*sum[floor$sqrt$n*n-x*x|x<-[0..n]] Counts points in the quadrant x>=0, y>0, multiplies by 4, adds 1 for the center point.
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f n=1+4*sum[floor$sqrt$n*n-x*x|x<-[0..n]] Counts points in the quadrant x>=0, y>0, multiplies by 4, adds 1 for the center point.