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

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

0,5

Comments

X-coordinates are given in A359216.

Links

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

Formula

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

Example

   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

Prog

(PARI) See Links section.

Crossrefs

Cf. A180714, A359058, A359216.

Sequence in context: A124478 A030353 A089617 * A359216 A274370 A128521

Adjacent sequences: A359214 A359215 A359216 * A359218 A359219 A359220

Keyword

sign,walk

Author

Hans G. Oberlack, Dec 21 2022

Status

approved