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

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, 1, 51, 95, 34, 62, 106, 10, 79, 117, 129, 21, 43, 141, 90, 1, 55, 68, 103, 31, 152, 13, 116, 80, 130, 165, 43, 180, 195, 1, 57, 92, 23, 142, 107, 209, 71, 225, 123

Offset: 1

Author

Peter Kagey, Jul 07 2017

Keywords

• nonn

Examples

			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.
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..3000
• Peter Kagey, Plot of the first 500 circles

Programs

Maple

A[1]:= 1:
for n from 2 to 100 do
  excl:= {}:
  for i from 1 to n-1 do
    if (i-n)^2 <= i+n or 4*n*i > ((i-n)^2 - (n+i))^2 then
      r:=  ceil(sqrt((sqrt(n)+sqrt(i))^2 - (n-i)^2))-1;
      excl:= excl union {$(A[i]-r) .. (A[i]+r)};
    fi
  od;
  A[n]:= min({$1..max(excl)+1} minus excl);
od:
seq(A[i],i=1..100); # Robert Israel, Jul 07 2017