A268038 List of y-coordinates of point moving in clockwise square spiral.

0, 0, -1, -1, -1, 0, 1, 1, 1, 1, 0, -1, -2, -2, -2, -2, -2, -1, 0, 1, 2, 2, 2, 2, 2, 2, 1, 0, -1, -2, -3, -3, -3, -3, -3, -3, -3, -2, -1, 0, 1, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 1, 0, -1, -2, -3, -4, -4, -4, -4, -4, -4, -4, -4, -4, -3, -2, -1, 0, 1, 2, 3, 4, 4, 4

Offset

1,13

Comments

This spiral, in either direction, is sometimes called the "Ulam spiral", but "square spiral" is a better name. (Ulam looked at the positions of the primes, but of course the spiral itself must be much older.) - N. J. A. Sloane, Jul 17 2018

Links

Peter Kagey, Table of n, a(n) for n = 1..10000

Visualization of spiral using Plot 2. - Hugo Pfoertner, May 29 2018

Example

Sequence gives y-coordinate of the n-th point of the following spiral:
.
  20--21--22--23--24--25
   |                   |
  19   6---7---8---9  26
   |   |           |   |
  18   5   0---1  10  27
   |   |       |   |   |
  17   4---3---2  11  28
   |               |   |
  16--15--14--13--12  29
                       |
  35--34--33--32--31--30

Mathematica

a[n_] := a[n] = If[n==0, 0, a[n-1] + Cos[Mod[Floor[Sqrt[4*(n-1) + 1]], 4]* Pi/2]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 11 2018, after Seppo Mustonen *)

Prog

(PARI)  L=1; d=-1;
for(r=1, 9, d=-d; k=floor(r/2)*d; for(j=1, L++, print1(k, ", ")); forstep(j=k-d, -floor((r+1)/2)*d+d, -d, print1(j, ", "))) \\ Hugo Pfoertner, Jul 28 2018

Crossrefs

A174344 gives sequence of x-coordinates.

This is the negative of A274923.

The (x,y) coordinates for a point sweeping a quadrant by antidiagonals are (A025581, A002262). - N. J. A. Sloane, Jul 17 2018

Sequence in context: A125088 A226456 A343642 * A274923 A249071 A231713

Adjacent sequences: A268035 A268036 A268037 * A268039 A268040 A268041

Keyword

sign,easy

Author

Peter Kagey, Jan 24 2016

Status

approved