A280317 Ordinate of points (x,y) of the square lattice such that x >= 0 and 0 <= y <= x, and ranked in order of increasing distance from the origin. Equidistant points are ranked in order of increasing ordinate.

0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 3, 2, 0, 3, 1, 2, 4, 3, 0, 1, 2, 4, 3, 0, 1, 5, 4, 2, 3, 5, 0, 1, 4, 2, 6, 3, 5, 4, 0, 1, 2, 6, 5, 3, 4, 7, 0, 6, 1, 2, 5, 3, 7, 4, 6, 0, 1, 2, 5, 8, 3, 7, 6, 4, 0, 1, 8, 5, 2, 7, 3, 6, 4, 9, 8, 0, 5, 1, 7, 2, 3, 6, 9, 4, 8, 7, 5, 0, 1, 2, 10, 9, 3, 6, 8, 4, 7, 5, 10

Offset: 1

Author

Robert G. Wilson v and Andres Cicuttin, Dec 31 2016

Keywords

• nonn

Examples

			a(12) = 3 since the twelfth point in distance from the origin is (3,3) at a distance of 3*sqrt(2) = 4.242640... whereas the eleventh is (4,1) at a distance of sqrt(17) = 4.12310... and the thirteenth is (4,2) at a distance of 2*sqrt(5) = 4.472113... .
The fourteenth and fifteenth points are respectively (5,0) and (4,3) and have the same distance 5 to the origin, but (5,0) has a smaller ordinate than (4,3), so a(14) = 0 and a(15) = 3.
		

Crossrefs

Cf. A280079.

Programs

Mathematica

xmax = 20; (* Maximum explorative abscissa *)
(* t are points in the triangle of vertices (0, 0), (0, max) and (xmax, xmax) *)
t = Flatten[Table[{x, y}, {x, 0, xmax}, {y, 0, x}], 1];
nmax = Floor[xmax^2/4] (* Safe limit for correctly sorted sequence *)
Transpose[SortBy[t, {#[[1]]^2 + #[[2]]^2 &, #[[2]] &}]][[2]][[1 ;;
   nmax]]