A305576 -id:A305576 - cp's OEIS frontend

A342562 List points (x,y,z) having integer coordinates, sorted first by R=x^2+y^2+z^2 and in case of ties, then by z and last by polar angle 0 <= phi < 2*Pi in a polar coordinate system. Sequence gives y-coordinates.

0, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 1, 1, -1, -1, 0, 1, 0, -1, 1, 1, -1, -1, 1, 1, -1, -1, 0, 0, 2, 0, -2, 0, 0, 1, 0, -1, 0, 2, 0, -2, 1, 2, 2, 1, -1, -2, -2, -1, 0, 2, 0, -2, 0, 1, 0, -1, 1, 1, -1, -1, 1, 2, 2, 1, -1, -2, -2, -1, 1, 2, 2, 1, -1, -2, -2, -1, 1, 1, -1, -1, 0, 2, 0, -2, 2, 2, -2, -2, 0, 2, 0, -2

Offset: 0

Hugo Pfoertner, Apr 27 2021

			See A342561.

Hugo Pfoertner, Table of n, a(n) for n = 0..10130

Cf. A305575, A305576, A342561 (x-coordinates), A342563 (z-coordinates).

A342562_vec=concat([[P[2] | P<-shell(n)] | n<-[0..7]]) \\ See A342561 for shell(). - M. F. Hasler, Apr 27 2021

A368121 A variant of A367150 with application of the distance minimization to an 8-fold symmetrically expanded sector between 0 and Pi/4 of the pair of rotated grids created by the strip bijection of A307110.

Original entry on oeis.org

0, 5, 6, 7, 8, 2, 3, 4, 1, 13, 15, 17, 19, 14, 10, 16, 11, 18, 12, 20, 9, 26, 27, 28, 25, 21, 22, 23, 24, 38, 39, 40, 41, 42, 43, 44, 37, 30, 31, 32, 33, 34, 35, 36, 29, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 61, 46, 47, 48, 45, 50, 51, 52, 53, 54, 55, 56, 49, 70, 82, 72, 73, 84, 75, 76, 86, 78, 79, 88, 69, 71, 103

Offset: 0

Hugo Pfoertner, Jan 05 2024

Hugo Pfoertner, Table of n, a(n) for n = 0..10000
Klaus Nagel, Symmetrized strip bijection, with mirrored repetition of sector 0.
Hugo Pfoertner, Distance minimization applied to symmetrized strip bijection, using sector between 0 and Pi/4.
Hugo Pfoertner, PARI program.

A305575, A305576 are used for enumeration of the grid points.

Cf. A307110, A367150, A368124, A368125, A368126.

PARI
```
\\ See Pfoertner link.
```

A368126 A variant of A367150 with application of the distance minimization to an 8-fold symmetrically expanded sector between Pi/4 and Pi/2 of the pair of rotated grids created by the strip bijection of A307110.

Original entry on oeis.org

0, 5, 6, 7, 8, 2, 3, 4, 1, 13, 15, 17, 19, 14, 10, 16, 11, 18, 12, 20, 9, 26, 27, 28, 25, 21, 22, 23, 24, 38, 39, 40, 41, 42, 43, 44, 37, 30, 31, 32, 33, 34, 35, 36, 29, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 61, 46, 47, 48, 45, 50, 51, 52, 53, 54, 55, 56, 49, 71, 72, 83, 74, 75, 85, 77, 78, 87, 80, 69, 81, 102, 73

Offset: 0

Hugo Pfoertner, Jan 05 2024

nonn

Hugo Pfoertner, Table of n, a(n) for n = 0..10000
Klaus Nagel, Symmetrized strip bijection, with mirrored repetition of sector 1.
Hugo Pfoertner, Distance minimization applied to symmetrized strip bijection, using sector between Pi/4 and Pi/2.
Hugo Pfoertner, PARI program.

A305575, A305576 are used to enumerate the grid points in rings of increasing radius.

Cf. A307110, A367150, A368121, A368129, A368130.

PARI
```
\\ See Pfoertner link.
```

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

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

sign

			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], ...

For the y coordinates see A283308.

Cf. A004018, A283303, A283304, A283305, A283306, A305575, A305576.

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

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

A308080 Index positions of the points with integer coordinates with primary sorting by radius and secondary sorting by polar angle in the counterclockwise square spiral described by A174344 and A274923.

Original entry on oeis.org

1, 2, 4, 6, 8, 3, 5, 7, 9, 11, 15, 19, 23, 12, 14, 16, 18, 20, 22, 24, 10, 13, 17, 21, 25, 28, 34, 40, 46, 29, 33, 35, 39, 41, 45, 47, 27, 30, 32, 36, 38, 42, 44, 48, 26, 53, 61, 69, 77, 54, 60, 62, 68, 70, 76, 78, 52, 31, 37, 43, 49, 55, 59, 63, 67, 71, 75, 79

Offset: 0

Hugo Pfoertner, May 11 2019

nonn

The lists of the ring-wise sorted points is provided in A305575 for x and A305576 for y.

The sequence is a permutation of the integers.

Hugo Pfoertner, Table of n, a(n) for n = 0..10000

Cf. A174344, A274923, A305575, A305576, A308081.

/* It is assumed that the files a305575 and a305576 contain the second column of the corresponding b-files omitting the initial 0 */
a305575=readvec(a305575); a305576=readvec(a305576);
a174344=vector(10000);L=0; d=1; n=0;
for(r=1, 100, d=-d; k=floor(r/2)*d; for(j=1, L++, a174344[n++]=k); forstep(j=k-d, -floor((r+1)/2)*d+d, -d, a174344[n++]=j));
a274923=vector(10100);L=1; d=1;n=0;
for(r=1, 100, d=-d; k=floor(r/2)*d; for(j=1, L++, a274923[n++]=k); forstep(j=k-d, -floor((r+1)/2)*d+d, -d, a274923[n++]=j));
findinspiral(i, j)={my(size=(2*max(abs(i), abs(j))+1)^2); forstep(k=size, 1, -1, if(i==a174344[k]&&j==a274923[k], return(k)))};
print1(findinspiral(0,0),", ");for(n=1,67,print1(findinspiral(a305575[n],a305576[n]),", "));

A357021 First coordinate x of points in the triangular lattice, sorted first by the distance from the origin and then by the circumferential angle phi restricted to the sector 0 <= phi < Pi/6. y is given in A357022.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 7, 8, 8, 8, 9, 9, 8, 9, 9, 10, 10, 10, 9, 10, 10, 11, 11, 11, 10, 11, 12, 12, 11, 12, 12, 11, 12, 13, 13, 12, 13, 13, 12, 13, 14, 14, 14, 14, 13, 14, 13, 15, 15, 14, 15, 15, 14, 15, 16, 16, 14, 16, 15

Offset: 1

Hugo Pfoertner, Sep 10 2022

nonn

The coordinates (x,y) are defined in an oblique coordinate system with an angle of 120 degrees between the axes, see e.g. A307012.
The distance from the origin is given by r = sqrt(x^2 - x*y + y^2), and the circumferential angle is phi = atan(sqrt(3)*y/(2*x - y)).
Using the pairs of terms of this sequence and of A357022(n) as grid indices in an infinite triangular lattice of one-ohm resistors leads to strictly increasing resistances against (0,0) (see A355585). This is similar to the role of A280079 and A280317 used as grid indices in the square lattice (see A355565).

			R is the resistance between a grid point (x,y) and (0,0) in an infinite triangular lattice of one-ohm resistors.
.
   n  x y  r^2   phi      R
              (degrees) (ohms)
   1  0 0   0          0.0000000000
   2  1 0   1    0.000 0.3333333333
   3  2 1   3   30.000 0.4359911242
   4  2 0   4    0.000 0.4613510850
   5  3 1   7   19.107 0.5132889542
   6  3 0   9    0.000 0.5362130198
   7  4 2  12   30.000 0.5627909282
   8  4 1  13   13.898 0.5700986140
   9  4 0  16    0.000 0.5891518971
  ...
  19  7 1  43    7.589 0.6800193341
  20  8 4  48   30.000 0.6901322715
  21  7 0  49    0.000 0.6920215369
  22  8 3  49   21.787 0.6920259223
  23  8 2  52   13.898 0.6974842443

Cf. A003136, A280079, A280317, A305575, A305576, A355565, A355585.

cp's OEIS Frontend

A342562 List points (x,y,z) having integer coordinates, sorted first by R=x^2+y^2+z^2 and in case of ties, then by z and last by polar angle 0 <= phi < 2*Pi in a polar coordinate system. Sequence gives y-coordinates.

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PARI

A368121 A variant of A367150 with application of the distance minimization to an 8-fold symmetrically expanded sector between 0 and Pi/4 of the pair of rotated grids created by the strip bijection of A307110.

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PARI

A368126 A variant of A367150 with application of the distance minimization to an 8-fold symmetrically expanded sector between Pi/4 and Pi/2 of the pair of rotated grids created by the strip bijection of A307110.

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PARI

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.

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Maple

PARI

A308080 Index positions of the points with integer coordinates with primary sorting by radius and secondary sorting by polar angle in the counterclockwise square spiral described by A174344 and A274923.

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PARI

A357021 First coordinate x of points in the triangular lattice, sorted first by the distance from the origin and then by the circumferential angle phi restricted to the sector 0 <= phi < Pi/6. y is given in A357022.

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