A357301 a(n) is the number of distinct radii of circles passing through at least three points in a square grid of n X n points.
0, 1, 7, 19, 48, 112, 212, 383, 641, 988, 1523, 2250, 3103, 4364, 5831, 7696, 9985, 12945, 16164, 20246, 24946, 30145, 36385, 43839, 51752, 61610, 72475, 84273, 97231, 112733
Offset: 1
Examples
a(2) = 1: only one possible circle with squared radius 1/2;
a(3) = 7: squared radii of possible circles are {1, 2, 1/2, 5/2, 5/4, 25/16, 25/18}.
Links
- Mathematics StackExchange, Number of circles in configuration, comment by Ed Pegg, Jan 09, 2013.
Programs
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PARI
\\ Function r2 determined using Mathematica \\ (ArcLength [CircleThrough[{{x1, y1}, {x2, y2}, {x3, y3}}]]/(2*Pi))^2 radii(n) = {my (R=List(), r2(x1,y1,x2,y2,x3,y3) = ((x3 + (x2^2*y1 - x3^2*y1 - x1^2*y2 + x3^2*y2 - y1^2*y2 + y1*y2^2 + x1^2*y3 - x2^2*y3 + y1^2*y3 - y2^2*y3 - y1*y3^2 + y2*y3^2)/(2*(-x2*y1 + x3*y1 + x1*y2 - x3*y2 - x1*y3 + x2*y3)))^2 + (y3 - (-2*(x2 - x3)*(x1^2 - x3^2 + y1^2 - y3^2) + 2*(x1 - x3)*(x2^2 - x3^2 + y2^2 - y3^2))/(-4*x2*y1 + 4*x3*y1 + 4*x1*y2 - 4*x3*y2 - 4*x1*y3 + 4*x2*y3))^2)); for(x1=1, n, for(y1=1, n, for(x2=1, x1, for(y2=1, n, for(x3=1, x2, for(y3=1, n, my (ax2=2*(x2-x1), ay2=2*(y2-y1), ax3=2*(x3-x1), ay3=2*(y3-y1), den=ax2*ay3 -ax3*ay2); if (den==0, next); listput (R, r2(x1,y1,x2,y2,x3,y3)))))))); Set(R)}; for (k=1,15, print1(#radii(k),", "))
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