The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A174344 List of x-coordinates of point moving in clockwise square spiral. 39
 0, 1, 1, 0, -1, -1, -1, 0, 1, 2, 2, 2, 2, 1, 0, -1, -2, -2, -2, -2, -2, -1, 0, 1, 2, 3, 3, 3, 3, 3, 3, 2, 1, 0, -1, -2, -3, -3, -3, -3, -3, -3, -3, -2, -1, 0, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 1, 0, -1, -2, -3, -4, -4, -4, -4, -4, -4, -4, -4, -4, -3, -2 (list; graph; refs; listen; history; text; internal format)
 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 S. 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(mod(floor(sqrt(4*(n-2)+1)),4)*Pi/2). For a corresponding formula for the y-coordinate, replace sin by 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 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 Cf. A180714. A268038 (or A274923) gives sequence of y-coordinates. The (x,y) coordinates for a point sweeping a quadrant by antidiagonals are (A025581, A002262). - N. J. A. Sloane, Jul 17 2018 See A296030 for the pairs (A174344(n), A274923(n)). - M. F. Hasler, Oct 20 2019 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 Sequence in context: A124752 A293730 A318722 * A049241 A321858 A334235 Adjacent sequences:  A174341 A174342 A174343 * A174345 A174346 A174347 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 23 21:27 EDT 2020. Contains 337315 sequences. (Running on oeis4.)