cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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.

Original entry on oeis.org

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, 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, 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
Offset: 1

Views

Author

N. J. A. Sloane, Mar 04 2017, following a suggestion from Ahmet Arduç

Keywords

Examples

			The first few points (listing [x^2+y^2,x,y]) are:
[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], ...

Links

Crossrefs

For the y coordinates see A283304.
See also A283305-A283308.

Programs

  • Maple
    L:=[];
M:=30;
for i from 0 to M do
for j from 0 to i do
L:=[op(L),[i^2+j^2,i,j]]; od: od:
t3:= sort(L,proc(a,b) evalb(a[1]<=b[1]); end);
t3x:=[seq(t3[i][2],i=1..100)]; # A283303
t3y:=[seq(t3[i][3],i=1..100)]; # A283304
  • Mathematica
    nt = 105; (* number of terms to produce *)
S[m_] := S[m] = Table[{x, y}, {y, 0, m}, {x, y, m}] // Flatten[#, 1]& // SortBy[#, {#.#&, #[[1]]&}]& // #[[All, 1]]& // PadRight[#, nt]&
S[m = 2];
S[m = 2 m];
While[S[m] =!= S[m/2], m = 2 m];
S[m] (* Jean-François Alcover, Mar 05 2023 *)

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.

Original entry on oeis.org

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, 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, 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
Offset: 1

Views

Author

N. J. A. Sloane, Mar 04 2017, following a suggestion from Ahmet Arduç

Keywords

Examples

			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], ...

Crossrefs

For the y coordinates see A283306.
See also A000925, A283303, A283304, A283307, A283308.

Programs

  • Maple
    L:=[];
M:=30;
for i from 0 to M do
for j from 0 to M do
L:=[op(L),[i^2+j^2,i,j]]; od: od:
t4:= sort(L,proc(a,b) evalb(a[1]<=b[1]); end);
t4x:=[seq(t4[i][2],i=1..100)]; # A283305
t4y:=[seq(t4[i][3],i=1..100)]; # A283306
  • Mathematica
    nt = 105; (* number of terms to produce *)
S[m_] := S[m] = Table[{x, y}, {x, 0, m}, {y, 0, m}] // Flatten[#, 1]& // SortBy[#, {#.#&, #[[1]]&}]& // #[[All, 1]]& // PadRight[#, nt]&;
S[m = 2];
S[m = 2m];
While[S[m] =!= S[m/2], m = 2m];
S[m] (* Jean-François Alcover, Mar 05 2023 *)
  • 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

A283308 List points (x,y) having integer coordinates, sorted first by x^2+y^2 and in case of ties, by x-coordinate and then by y-coordinate. Sequence gives y-coordinates.

Original entry on oeis.org

0, 0, -1, 1, 0, -1, 1, -1, 1, 0, -2, 2, 0, -1, 1, -2, 2, -2, 2, -1, 1, -2, 2, -2, 2, 0, -3, 3, 0, -1, 1, -3, 3, -3, 3, -1, 1, -2, 2, -3, 3, -3, 3, -2, 2, 0, -4, 4, 0, -1, 1, -4, 4, -4, 4, -1, 1, -3, 3, -3, 3, -2, 2, -4, 4, -4, 4, -2, 2, 0, -3, 3, -4, 4, -5, 5, -4, 4, -3, 3, 0, -1, 1, -5, 5, -5, 5
Offset: 1

Views

Author

N. J. A. Sloane, Mar 04 2017, following a suggestion from Ahmet Arduç

Keywords

Examples

			The first few points (listing [x^2+y^2,x,y]) are: [0, 0, 0], [1, -1, 0], [1, 0, -1], [1, 0, 1], [1, 1, 0], [2, -1, -1], [2, -1, 1], [2, 1, -1], [2, 1, 1], [4, -2, 0], [4, 0, -2], [4, 0, 2], [4, 2, 0], [5, -2, -1], [5, -2, 1], [5, -1, -2], [5, -1, 2], [5, 1, -2], [5, 1, 2], [5, 2, -1], [5, 2, 1], [8, -2, -2], [8, -2, 2], [8, 2, -2], ...

Crossrefs

For the x coordinates see A283307.
Cf. A004018, A228108, A283303, A283304, A283305, A283306, A305575, A305576.

Programs

  • Maple
    L:=[];
M:=30;
for i from -M to M do
for j from -M to M do
L:=[op(L),[i^2+j^2,i,j]]; od: od:
t6:= sort(L,proc(a,b) evalb(a[1]<=b[1]); end);
t6x:=[seq(t6[i][2],i=1..100)]; # A283307
t6y:=[seq(t6[i][3],i=1..100)]; # A283308
  • PARI
    rs(t)=round(sqrt(abs(t)));pt(t)=print1(rs(t)*sign(t),", ");for(r2=0,26,xm=rs(r2);for(x=-xm,xm,y2=r2-x^2;if(issquare(y2),if(y2==0,pt(0),pt(-y2);pt(y2))))) \\ Hugo Pfoertner, Jun 18 2018

A283306 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 y-coordinates.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Mar 04 2017, following a suggestion from Ahmet Arduç

Keywords

Examples

			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], ...

Crossrefs

For the x coordinates see A283305.
See also A283303, A283304.

Programs

  • Maple
    L:=[];
M:=30;
for i from 0 to M do
for j from 0 to M do
L:=[op(L),[i^2+j^2,i,j]]; od: od:
t4:= sort(L,proc(a,b) evalb(a[1]<=b[1]); end);
t4x:=[seq(t4[i][2],i=1..100)]; # A283305
t4y:=[seq(t4[i][3],i=1..100)]; # A283306
  • Mathematica
    nt = 105; (* number of terms to produce *)
S[m_] := S[m] = Table[{x, y}, {x, 0, m}, {y, 0, m}] // Flatten[#, 1]& // SortBy[#, {#.#&, #[[1]]&}]& // #[[All, 2]]& // PadRight[#, nt]&
S[m = 2];
S[m = 2m];
While[S[m] =!= S[m/2], m = 2m];
S[m] (* Jean-François Alcover, Mar 05 2023 *)
  • PARI
    for(r2=0,113,for(x=0,round(sqrt(r2)),y2=r2-x^2; if(issquare(y2), print1(round(sqrt(y2)),", ")))) \\ Hugo Pfoertner, Jun 18 2018

A283307 List points (x,y) having integer coordinates, sorted first by x^2+y^2 and in case of ties, by x-coordinate and then by y-coordinate. Sequence gives x-coordinates.

Original entry on oeis.org

0, -1, 0, 0, 1, -1, -1, 1, 1, -2, 0, 0, 2, -2, -2, -1, -1, 1, 1, 2, 2, -2, -2, 2, 2, -3, 0, 0, 3, -3, -3, -1, -1, 1, 1, 3, 3, -3, -3, -2, -2, 2, 2, 3, 3, -4, 0, 0, 4, -4, -4, -1, -1, 1, 1, 4, 4, -3, -3, 3, 3, -4, -4, -2, -2, 2, 2, 4, 4, -5, -4, -4, -3, -3, 0, 0, 3, 3, 4, 4, 5, -5, -5, -1, -1, 1
Offset: 1

Views

Author

N. J. A. Sloane, Mar 04 2017, following a suggestion from Ahmet Arduç

Keywords

Examples

			The first few points (listing [x^2+y^2,x,y]) are: [0, 0, 0], [1, -1, 0], [1, 0, -1], [1, 0, 1], [1, 1, 0], [2, -1, -1], [2, -1, 1], [2, 1, -1], [2, 1, 1], [4, -2, 0], [4, 0, -2], [4, 0, 2], [4, 2, 0], [5, -2, -1], [5, -2, 1], [5, -1, -2], [5, -1, 2], [5, 1, -2], [5, 1, 2], [5, 2, -1], [5, 2, 1], [8, -2, -2], [8, -2, 2], [8, 2, -2], ...

Crossrefs

For the y coordinates see A283308.
Cf. A004018, A283303, A283304, A283305, A283306, A305575, A305576.

Programs

  • Maple
    L:=[];
M:=30;
for i from -M to M do
for j from -M to M do
L:=[op(L),[i^2+j^2,i,j]]; od: od:
t6:= sort(L,proc(a,b) evalb(a[1]<=b[1]); end);
t6x:=[seq(t6[i][2],i=1..100)]; # A283307
t6y:=[seq(t6[i][3],i=1..100)]; # A283308
  • PARI
    pt(t)=print1(t,", ");for(r2=0,26,xm=round(sqrt(r2));for(x=-xm,xm,y2=r2-x^2;if(issquare(y2),if(y2!=0,pt(x));pt(x)))) \\ Hugo Pfoertner, Jun 18 2018

A229140 Smallest k such that k^2 + l^2 = n-th number expressible as sum of two squares (A001481).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 3, 2, 0, 1, 2, 4, 3, 0, 1, 2, 4, 3, 0, 1, 4, 2, 3, 5, 0, 1, 2, 6, 3, 5, 4, 0, 1, 2, 5, 3, 4, 7, 0, 1, 2, 5, 3, 7, 4, 6, 0, 1, 2, 8, 3, 6, 4, 0, 1, 5, 2, 7, 3, 6, 4, 9, 8, 0, 1, 2, 3, 6, 9, 4, 7, 5, 0, 1, 2, 9, 3, 8, 4, 7, 5, 0
Offset: 1

Views

Author

Ralf Stephan, Sep 15 2013

Keywords

Comments

Conjecture: the values between two zeros are always distinct from each other.

Examples

			The 6th number expressible as sum of two squares A001481(6) = 8 = 2^2 + 2^2, so a(6)=2.

Crossrefs

Cf. A001481, A385236 (largest k), A385237, A283303, A283304.

Programs

  • PARI
    for(n=0, 300, s=sqrtint(n); forstep(i=s, 0, -1, if(issquare(n-i*i), print1(sqrtint(n-i*i), ", "); break))); \\ shift corrected by Michel Marcus, Jul 08 2025

Formula

a(n) = 0 if A001481(n) is square.
a(n) = sqrt(A001481(n)-A385236(n)^2). - Zhuorui He, Jul 08 2025

A385236 Largest x such that x^2+y^2 = A001481(n), x and y are nonnegative integers.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 3, 4, 5, 5, 5, 4, 5, 6, 6, 6, 5, 6, 7, 7, 6, 7, 7, 6, 8, 8, 8, 6, 8, 7, 8, 9, 9, 9, 8, 9, 9, 7, 10, 10, 10, 9, 10, 8, 10, 9, 11, 11, 11, 8, 11, 10, 11, 12, 12, 11, 12, 10, 12, 11, 12, 9, 10, 13, 13, 13, 13, 12, 10, 13, 12, 13, 14, 14, 14, 11, 14, 12, 14, 13, 14, 15
Offset: 1

Views

Author

Zhuorui He, Jul 08 2025

Keywords

Comments

A229140(n) gives smallest x such that x^2+y^2 = A001481(n), x and y are nonnegative integers.

Examples

			For n=9, A001481(9)=13=2^2+3^2, so A229140(9)=2 and a(9)=3.
For n=14, A001481(14)=25=3^2+4^2=0^2+5^2, so A229140(14)=0 and a(14)=5.

Crossrefs

Cf. A001481, A229140, A283303, A283304.

Programs

  • PARI
    for(n=0, 300, s=sqrtint(n); forstep(i=s, 0, -1, if(issquare(n-i*i), print1(i, ", "); break)))

Formula

a(n) = sqrt(A001481(n)) if A001481(n) is square.
a(n) = sqrt(A001481(n)-A229140(n)^2).
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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