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A283303 List points (x,y) having integer coordinates with x >= y >= 0, sorted first by x^2+y^2 and in case of a tie, by x-coordinate. Sequence gives x-coordinates.

%I A283303 #25 Jul 09 2025 04:42:41
%S A283303 0,1,1,2,2,2,3,3,3,4,4,3,4,4,5,5,5,4,5,6,6,6,5,6,7,5,7,6,7,7,6,8,7,8,
%T A283303 8,6,8,7,8,9,9,7,9,8,9,9,7,8,10,10,10,9,10,8,10,9,11,11,10,11,8,9,11,
%U A283303 10,11,12,9,12,11,12,10,12,11,12,9,10,12,13,11,13,13,13,12,10,11,13,12,13,14
%N A283303 List points (x,y) having integer coordinates with x >= y >= 0, sorted first by x^2+y^2 and in case of a tie, by x-coordinate. Sequence gives x-coordinates.
%H A283303 Hugo Pfoertner, <a href="/A283303/b283303.txt">Table of n, a(n) for n = 1..10000</a>
%e A283303 The first few points (listing [x^2+y^2,x,y]) are:
%e A283303 [0, 0, 0], [1, 1, 0], [2, 1, 1], [4, 2, 0], [5, 2, 1], [8, 2, 2], [9, 3, 0], [10, 3, 1], [13, 3, 2], [16, 4, 0], [17, 4, 1], [18, 3, 3], [20, 4, 2], [25, 4, 3], [25, 5, 0], [26, 5, 1], [29, 5, 2], [32, 4, 4], [34, 5, 3], [36, 6, 0], [37, 6, 1], [40, 6, 2], [41, 5, 4], [45, 6, 3], [49, 7, 0], ...
%p A283303 L:=[];
%p A283303 M:=30;
%p A283303 for i from 0 to M do
%p A283303 for j from 0 to i do
%p A283303 L:=[op(L),[i^2+j^2,i,j]]; od: od:
%p A283303 t3:= sort(L,proc(a,b) evalb(a[1]<=b[1]); end);
%p A283303 t3x:=[seq(t3[i][2],i=1..100)]; # A283303
%p A283303 t3y:=[seq(t3[i][3],i=1..100)]; # A283304
%t A283303 nt = 105; (* number of terms to produce *)
%t A283303 S[m_] := S[m] = Table[{x, y}, {y, 0, m}, {x, y, m}] // Flatten[#, 1]& // SortBy[#, {#.#&, #[[1]]&}]& // #[[All, 1]]& // PadRight[#, nt]&
%t A283303 S[m = 2];
%t A283303 S[m = 2 m];
%t A283303 While[S[m] =!= S[m/2], m = 2 m];
%t A283303 S[m] (* _Jean-François Alcover_, Mar 05 2023 *)
%Y A283303 For the y coordinates see A283304.
%Y A283303 See also A283305-A283308.
%K A283303 nonn
%O A283303 1,4
%A A283303 _N. J. A. Sloane_, Mar 04 2017, following a suggestion from _Ahmet Arduç_