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A289523 Lexicographically earliest sequence of positive integers such that no circles centered at (n, a(n)) with radius sqrt(n) overlap.

%I A289523 #12 Jul 08 2017 00:48:34
%S A289523 1,4,7,1,11,16,5,21,27,34,10,1,41,17,49,25,57,6,33,66,43,14,75,85,24,
%T A289523 1,51,95,34,62,106,10,79,117,129,21,43,141,90,1,55,68,103,31,152,13,
%U A289523 116,80,130,165,43,180,195,1,57,92,23,142,107,209,71,225,123
%N A289523 Lexicographically earliest sequence of positive integers such that no circles centered at (n, a(n)) with radius sqrt(n) overlap.
%H A289523 Peter Kagey, <a href="/A289523/b289523.txt">Table of n, a(n) for n = 1..3000</a>
%H A289523 Peter Kagey, <a href="/A289523/a289523_1.png">Plot of the first 500 circles</a>
%e A289523 For n = 3, a(3) = 7 because a circle centered at (3, 1) with radius sqrt(3) intersects the circle centered at (1, 1) with radius sqrt(1); a circle centered at (3, k) with radius sqrt(3) intersects the circle centered at (2, 4) with radius sqrt(2), for 2 <= k <= 6; therefore the circle centered at (3, 7) is the circle with the least y-coordinate that does not intersect any of the existing circles.
%p A289523 A[1]:= 1:
%p A289523 for n from 2 to 100 do
%p A289523   excl:= {}:
%p A289523   for i from 1 to n-1 do
%p A289523     if (i-n)^2 <= i+n or 4*n*i > ((i-n)^2 - (n+i))^2 then
%p A289523       r:=  ceil(sqrt((sqrt(n)+sqrt(i))^2 - (n-i)^2))-1;
%p A289523       excl:= excl union {$(A[i]-r) .. (A[i]+r)};
%p A289523     fi
%p A289523   od;
%p A289523   A[n]:= min({$1..max(excl)+1} minus excl);
%p A289523 od:
%p A289523 seq(A[i],i=1..100); # _Robert Israel_, Jul 07 2017
%K A289523 nonn
%O A289523 1,2
%A A289523 _Peter Kagey_, Jul 07 2017