cp's OEIS Frontend

A280079 Abscissa 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.

%I A280079 #41 Aug 28 2022 14:59:42
%S A280079 0,1,1,2,2,2,3,3,3,4,4,3,4,5,4,5,5,4,5,6,6,6,5,6,7,7,5,6,7,7,6,8,8,7,
%T A280079 8,6,8,7,8,9,9,9,7,8,9,9,7,10,8,10,10,9,10,8,10,9,11,11,11,10,8,11,9,
%U A280079 10,11,12,12,9,11,12,10,12,11,12,9,10,13,12,13,11,13,13,12,10,13,11,12,13,14,14,14,10,11,14,13,12,14,13,14,11
%N A280079 Abscissa 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.
%C A280079 The points belong to the first half of the first quadrant, and in order are (0,0), (1,0), (1,1), (2,0), (2,1), (2,2), (3,0), (3,1), (3,2), (4,0), (4,1), (3,3), (4,2), (5,0), (4,3), (5,1), (5,2), etc.
%e A280079 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... .
%e A280079 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) = 5 and a(15) = 4.
%t A280079 xmax = 20; (* Maximum explorative abscissa *)
%t A280079 (* t are points in the triangle of vertices (0,0), (0,max) and (xmax,xmax) *)
%t A280079 t = Flatten[Table[{x, y}, {x, 0, xmax}, {y, 0, x}], 1];
%t A280079 nmax = Floor[xmax^2/4] (* Safe limit for correctly sorted sequence *)
%t A280079 Transpose[SortBy[t, {#[[1]]^2 + #[[2]]^2 &, #[[2]] &}]][[1]][[1 ;;
%t A280079    nmax]]
%Y A280079 A280317 are the corresponding ordinates y.
%K A280079 nonn
%O A280079 1,4
%A A280079 _Robert G. Wilson v_ and _Andres Cicuttin_, Dec 25 2016