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A283305 List points (x,y) having integer coordinates with x >= 0, y >= 0, sorted first by x^2+y^2 and in case of a tie, by x-coordinate. Sequence gives x-coordinates.

%I A283305 #23 Jul 09 2025 04:42:54
%S A283305 0,0,1,1,0,2,1,2,2,0,3,1,3,2,3,0,4,1,4,3,2,4,0,3,4,5,1,5,2,5,4,3,5,0,
%T A283305 6,1,6,2,6,4,5,3,6,0,7,1,5,7,4,6,2,7,3,7,5,6,0,8,1,4,7,8,2,8,6,3,8,5,
%U A283305 7,4,8,0,9,1,9,2,6,7,9,5,8,3,9,4,9,7,0,6,8,10,1,10,2,10,5,9,3,10,7,8
%N A283305 List points (x,y) having integer coordinates with x >= 0, y >= 0, sorted first by x^2+y^2 and in case of a tie, by x-coordinate. Sequence gives x-coordinates.
%e A283305 The first few points (listing [x^2+y^2,x,y]) are: [0, 0, 0], [1, 0, 1], [1, 1, 0], [2, 1, 1], [4, 0, 2], [4, 2, 0], [5, 1, 2], [5, 2, 1], [8, 2, 2], [9, 0, 3], [9, 3, 0], [10, 1, 3], [10, 3, 1], [13, 2, 3], [13, 3, 2], [16, 0, 4], [16, 4, 0], [17, 1, 4], [17, 4, 1], [18, 3, 3], [20, 2, 4], [20, 4, 2], [25, 0, 5], [25, 3, 4], [25, 4, 3], ...
%p A283305 L:=[];
%p A283305 M:=30;
%p A283305 for i from 0 to M do
%p A283305 for j from 0 to M do
%p A283305 L:=[op(L),[i^2+j^2,i,j]]; od: od:
%p A283305 t4:= sort(L,proc(a,b) evalb(a[1]<=b[1]); end);
%p A283305 t4x:=[seq(t4[i][2],i=1..100)]; # A283305
%p A283305 t4y:=[seq(t4[i][3],i=1..100)]; # A283306
%t A283305 nt = 105; (* number of terms to produce *)
%t A283305 S[m_] := S[m] = Table[{x, y}, {x, 0, m}, {y, 0, m}] // Flatten[#, 1]& // SortBy[#, {#.#&, #[[1]]&}]& // #[[All, 1]]& // PadRight[#, nt]&;
%t A283305 S[m = 2];
%t A283305 S[m = 2m];
%t A283305 While[S[m] =!= S[m/2], m = 2m];
%t A283305 S[m] (* _Jean-François Alcover_, Mar 05 2023 *)
%o A283305 (PARI) for(r2=0,113,for(x=0,round(sqrt(r2)),y2=r2-x^2;if(issquare(y2),print1(x,", ")))) \\ _Hugo Pfoertner_, Jun 18 2018
%Y A283305 For the y coordinates see A283306.
%Y A283305 See also A000925, A283303, A283304, A283307, A283308.
%K A283305 nonn
%O A283305 1,6
%A A283305 _N. J. A. Sloane_, Mar 04 2017, following a suggestion from _Ahmet Arduç_