A359058 -id:A359058 - cp's OEIS frontend

A359216 X-coordinates of a point moving in a counterclockwise undulating spiral in a square grid.

0, 1, 1, 0, 0, -1, -1, -2, -2, -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 2, 2, 1, 1, 0, 0, -1, -1, -2, -2, -3, -3, -4, -4, -3, -3, -2, -2, -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 0, 0, -1, -1, -2, -2, -3, -3, -4, -4, -5, -5, -6, -6, -5, -5, -4

Offset: 0

Hans G. Oberlack, Dec 21 2022

Y-coordinates are given in A359217.
The undulating spiral is
   y ^
     |
   4 |             25--24
     |              |   |
   3 |         27--26  23--22
     |          |           |
   2 |     29--28   5---4  21--20
     |      |       |   |       |
   1 | 31--30   7---6   3---2  19--18
     |  |       |           |       |
   0 | 32--33   8---9   0---1  16--17
     |      |       |           |       |
  -1 |     34--35  10--11  14--15  46--47
     |          |       |   |       |
  -2 |         36--37  12--13  44--45
     |              |           |
  -3 |             38--39  42--43
     |                  |   |
  -4 |                 40--41
     +------------------------------------>
       -4  -3  -2  -1   0   1   2   3   4 x

Rémy Sigrist, Table of n, a(n) for n = 0..10081
Hans G. Oberlack, Counterclockwise undulating spiral in a square grid
Rémy Sigrist, PARI program
Index entries for sequences related to coordinates of 2D curves

Cf. A329116, A359058, A359217.

PARI
```
See Links section.
```

a(2*n) = A329116(n). - Rémy Sigrist, Apr 01 2023

A359217 Y-coordinates of a point moving along a counterclockwise undulating spiral on a square grid.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 1, 1, 0, 0, -1, -1, -2, -2, -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 3, 3, 2, 2, 1, 1, 0, 0, -1, -1, -2, -2, -3, -3, -4, -4, -3, -3, -2, -2, -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 0, 0, -1, -1

Offset: 0

Hans G. Oberlack, Dec 21 2022

X-coordinates are given in A359216.

			   y ^
     |
   4 |             25--24
     |              |   |
   3 |         27--26  23--22
     |          |           |
   2 |     29--28   5---4  21--20
     |      |       |   |       |
   1 | 31--30   7---6   3---2  19--18
     |  |       |           |       |
   0 | 32--33   8---9   0---1  16--17
     |      |       |           |       |
  -1 |     34--35  10--11  14--15  46--47
     |          |       |   |       |
  -2 |         36--37  12--13  44--45
     |              |           |
  -3 |             38--39  42--43
     |                  |   |
  -4 |                 40--41
     +------------------------------------>
       -4  -3  -2  -1   0   1   2   3   4 x

Rémy Sigrist, Table of n, a(n) for n = 0..10081
Hans G. Oberlack, Counterclockwise undulating spiral in a square grid
Rémy Sigrist, PARI program
Index entries for sequences related to coordinates of 2D curves

Cf. A180714, A359058, A359216.

PARI
```
See Links section.
```

Conjecture: a(n) = T10 + T15 + T20 + T21 where
  T1  = floor(n/16);
  T2  = sqrt(2*T1 + 1/4);
  T3  = floor(T2 - 1/2);
  T4  = n - T3*(T3+1)*16/2;
  T5  = (T3+1)*16;
  T6  = T4 + (3/4)*T5 - 1;
  T7  = T6/T5;
  T8  = floor(T7);
  T9  = 1 - T8;
  T10 = T9 - floor(T4/2);
  T11 = T4 + (2/4)*T5 - 1;
  T12 = T11/T5;
  T13 = floor(T12);
  T14 = T8 - T13;
  T15 = T14*floor((T5 - T11)/2);
  T16 = T4 + (1/4)*T5 - 1;
  T17 = T16/T5;
  T18 = floor(T17);
  T19 = T13 - T18;
  T20 = -T19*floor((T4 - T5/2)/2);
  T21 = -T18*floor((T5 - T4 + 1)/2).
a(2*n) = A180714(n). - Rémy Sigrist, Apr 01 2023

A367914 Movement sequence in the counter-clockwise undulating spiral, whereby 1, 2, 3, 4 represent moves to the right, down, left and up.

Original entry on oeis.org

1, 4, 3, 4, 3, 2, 3, 2, 1, 2, 1, 2, 1, 4, 1, 4, 1, 4, 3, 4, 3, 4, 3, 4, 3, 2, 3, 2, 3, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4

Offset: 1

Hans G. Oberlack, Dec 04 2023

  y ^
    |
  4 |              2---3
    |              |   |
  3 |          2---3   4---3
    |          |           |
  2 |      2---3   2---3   4---3
    |      |       |   |       |
  1 |  2---3   2---3   4---3   4---3
    |  |       |           |       |
  0 |  1---2   1---2   1---4   1---4
    |      |       |           |       |
 -1 |      1---2   1-- 2   1---4   1---4
    |          |       |   |       |
 -2 |          1---2   1---4   1---4
    |              |           |
 -3 |              1---2   1---4
    |                  |   |
 -4 |                  1---4
    +------------------------------------>
      -4  -3  -2  -1   0   1   2   3   4 x

Cf. A359058, A359216, A359217, A063826.

a(k1)=1 with k1=i^2*8+i*0+2*j+1  with i,j >= 0 and j<=4i.
a(k2)=2 with k2=i^2*8+i*12+2*j+6 with i,j >= 0 and j<=4*i+3.
a(k3)=3 with k3=i^2*8+i*8+2*j+3 with  i,j >= 0 and j<=4*i+2.
a(k4)=4 with k4=i^2*8+i*4+2*j+2 with  i,j >= 0 and j<=4*i+1.

cp's OEIS Frontend

A359216 X-coordinates of a point moving in a counterclockwise undulating spiral in a square grid.

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A359217 Y-coordinates of a point moving along a counterclockwise undulating spiral on a square grid.

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Links

Crossrefs

Programs

PARI

Formula

A367914 Movement sequence in the counter-clockwise undulating spiral, whereby 1, 2, 3, 4 represent moves to the right, down, left and up.

Views

Author

Keywords

Comments

Crossrefs

Formula