OFFSET
1,10
COMMENTS
Also, list of x-coordinates of point moving in counterclockwise square spiral.
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
Graham, Knuth and Patashnik give an exercise and answer on mapping n to square spiral x,y coordinates, and back x,y to n. They start 0 at the origin and first segment North so their y(n) is a(n+1). In their table of sides, it can be convenient to take n-4*k^2 so the ranges split at -m, 0, m. - Kevin Ryde, Sep 16 2019
REFERENCES
Ronald L. Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics, Addison-Wesley, 1989, chapter 3, Integer Functions, exercise 40 page 99 and answer page 498.
LINKS
Peter Kagey, Table of n, a(n) for n = 1..10000
Seppo Mustonen, Ulam spiral in color [Interactive web page]
Seppo Mustonen, Ulam spiral in color [Local copy of a snapshot of the page]
Hugo Pfoertner, Visualization of spiral using Plot 2, May 29 2018
N. J. A. Sloane, Ulam spiral in color.
Aaron Snook, Augmented Integer Linear Recurrences, Thesis, 2012.
FORMULA
a(1) = 0, a(n) = a(n-1) + sin(floor(sqrt(4n-7))*Pi/2). For a corresponding formula for the y-coordinate, replace sin with cos. - Seppo Mustonen, Aug 21 2010 with correction by Peter Kagey, Jan 24 2016
EXAMPLE
Here is the beginning of the clockwise square spiral. Sequence gives x-coordinate of the n-th point.
.
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--32--30
.
Given the offset equal to 1, a(n) gives the x-coordinate of the point labeled n-1 in the above drawing. - M. F. Hasler, Nov 03 2019
MAPLE
fx:=proc(n) option remember; local k; if n=1 then 0 else
k:=floor(sqrt(4*(n-2)+1)) mod 4;
fx(n-1) + sin(k*Pi/2); fi; end;
[seq(fx(n), n=1..120)]; # Based on Seppo Mustonen's formula. - N. J. A. Sloane, Jul 11 2016
MATHEMATICA
a[n_]:=a[n]=If[n==0, 0, a[n-1]+Sin[Mod[Floor[Sqrt[4*(n-1)+1]], 4]*Pi/2]]; Table[a[n], {n, 0, 50}] (* Seppo Mustonen, Aug 21 2010 *)
PROG
(PARI) L=0; 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
(PARI) a(n) = n--; my(m=sqrtint(n), k=ceil(m/2)); n -= 4*k^2; if(n<0, if(n<-m, k, -k-n), if(n<m, -k, n-3*k)); \\ Kevin Ryde, Sep 16 2019
(PARI) apply( A174344(n)={my(m=sqrtint(n-=1), k=m\/2); if(n < 4*k^2-m, k, 0 > n -= 4*k^2, -k-n, n < m, -k, n-3*k)}, [1..99]) \\ M. F. Hasler, Oct 20 2019
(Julia)
function SquareSpiral(len)
x, y, i, j, N, n, c = 0, 0, 0, 0, 0, 0, 0
for k in 0:len-1
print("$x, ") # or print("$y, ") for A268038.
if n == 0
c += 1; c > 3 && (c = 0)
c == 0 && (i = 0; j = 1)
c == 1 && (i = 1; j = 0)
c == 2 && (i = 0; j = -1)
c == 3 && (i = -1; j = 0)
c in [1, 3] && (N += 1)
n = N
end
n -= 1
x, y = x + i, y + j
end end
SquareSpiral(75) # Peter Luschny, May 05 2019
CROSSREFS
The (x,y) coordinates for a point sweeping a quadrant by antidiagonals are (A025581, A002262). - N. J. A. Sloane, Jul 17 2018
The diagonal rays are: A002939 (2*n*(2*n-1): 0, 2, 12, 30, ...), A016742 = (4n^2: 0, 4, 16, 36, ...), A002943 (2n(2n+1): 0, 6, 20, 42, ...), A033996 = (4n(n+1): 0, 8, 24, 48, ...). - M. F. Hasler, Oct 31 2019
KEYWORD
sign
AUTHOR
Nikolas Garofil (nikolas(AT)garofil.be), Mar 16 2010
EXTENSIONS
Link corrected by Seppo Mustonen, Sep 05 2010
Definition clarified by N. J. A. Sloane, Dec 20 2012
STATUS
approved