A180714 -id:A180714 - cp's OEIS frontend

A174344 List of x-coordinates of point moving in clockwise square spiral.

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

Offset: 1

Nikolas Garofil (nikolas(AT)garofil.be), Mar 16 2010

sign

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

			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

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.

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.
Index entries for sequences related to coordinates of 2D curves

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

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

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

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 *)

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

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(nKevin Ryde, Sep 16 2019

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

# Based on Kevin Ryde's PARI script
import math
def A174344(n):
    n -= 1
    m = math.isqrt(n)
    k = math.ceil(m/2)
    n -= 4*k*k
    if n < 0: return k if n < -m else -k-n
    return -k if n < m else n-3*k # David Radcliffe, Aug 04 2025

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

a(n) = A010751(A037458(n-1)) for n>1. - William McCarty, Jul 29 2021

Link corrected by Seppo Mustonen, Sep 05 2010

Definition clarified by N. J. A. Sloane, Dec 20 2012

A296030 Pairs of coordinates for successive integers in the square spiral (counterclockwise).

Original entry on oeis.org

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

Offset: 1

Benjamin Mintz, Dec 03 2017

The spiral is also called the Ulam spiral, cf. A174344, A274923 (x and y coordinates). - M. F. Hasler, Oct 20 2019
The n-th positive integer occupies the point whose x- and y-coordinates are represented in the sequence by a(2n-1) and a(2n), respectively. - Robert G. Wilson v, Dec 03 2017
From Robert G. Wilson v, Dec 05 2017: (Start)
The cover of the March 1964 issue of Scientific American (see link) depicts the Ulam Spiral with a heavy black line separating the numbers from their non-sequential neighbors. The pairs of coordinates for the points on this line, assuming it starts at the origin, form this sequence, negated.
The first number which has an abscissa value of k beginning at 0: 1, 2, 10, 26, 50, 82, 122, 170, 226, 290, 362, 442, 530, 626, 730, 842, 962, ...; g.f.: -(x^3 +7x^2 -x +1)/(x-1)^3;
The first number which has an abscissa value of -k beginning at 0: 1, 5, 17, 37, 65, 101, 145, 197, 257, 325, 401, 485, 577, 677, 785, 901, ...; g.f.: -(5x^2 +2x +1)/(x-1)^3;
The first number which has an ordinate value of k beginning at 0: 1, 3, 13, 31, 57, 91, 133, 183, 241, 307, 381, 463, 553, 651, 757, 871, 993, ...; g.f.: -(7x^2+1)/(x-1)^3;
The first number which has an ordinate value of -k beginning at 0: 1, 7, 21, 43, 73, 111, 157, 211, 273, 343, 421, 507, 601, 703, 813, 931, ...; g.f.: -(3x^2+4x+1)/(x-1)^3;
The union of the four sequences above is A033638.
(End)
Sequences A174344, A268038 and A274923 start with the integer 0 at the origin (0,0). One might then prefer offset 0 as to have (a(2n), a(2n+1)) as coordinates of the integer n. - M. F. Hasler, Oct 20 2019
This sequence can be read as an infinite table with 2 columns, where row n gives the x- and y-coordinate of the n-th point on the spiral. If the point at the origin has number 0, then the points with coordinates (n,n), (-n,n), (n,-n) and (n,-n) have numbers given by A002939(n) = 2n(2n-1): (0, 2, 12, 30, ...), A016742(n) = 4n^2: (0, 4, 16, 36, ...), A002943(n) = 2n(2n+1): (0, 6, 20, 42, ...) and A033996(n) = 4n(n+1): (0, 8, 24, 48, ...), respectively. - M. F. Hasler, Nov 02 2019

			The integer 1 occupies the initial position, so its coordinates are {0,0}; therefore a(1)=0 and a(2)=0.
The integer 2 occupies the position immediately to the right of 1, so its coordinates are {1,0}.
The integer 3 occupies the position immediately above 2, so its coordinates are {1,1}; etc.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 935.

Benjamin Mintz, Table of n, a(n) for n = 1..100000
BackIssues.com, Scientific American March 1964 back issue
Scientific American, March 1964 cover
Wikipedia, Ulam Spiral.

Cf. A033638, A063826, A174344, A180714, A268038, A274923.

Cf. Diagonal rays (+-n,+-n): A002939 (2n(2n-1): 0, 2, 12, 30, ...: NE), A016742 (4n^2: 0, 4, 16, 36, ...: NW), A002943 (2n(2n+1): 0, 6, 20, 42, ...: SW) and A033996 (4n(n+1): 0, 8, 24, 48, ...: SE).

f[n_] := Block[{k = Ceiling[(Sqrt[n] - 1)/2], m, t}, t = 2k +1; m = t^2; t--; If[n >= m - t, {k -(m - n), -k}, m -= t; If[n >= m - t, {-k, -k +(m - n)}, m -= t; If[n >= m - t, {-k +(m - n), k}, {k, k -(m - n - t)}]]]]; Array[f, 40] // Flatten (* Robert G. Wilson v, Dec 04 2017 *)
f[n_] := Block[{k = Mod[ Floor[ Sqrt[4 If[OddQ@ n, (n + 1)/2 - 2, (n/2 - 2)] + 1]], 4]}, f[n - 2] + If[OddQ@ n, Sin[k*Pi/2], -Cos[k*Pi/2]]]; f[1] = f[2] = 0; Array[f, 90] (* Robert G. Wilson v, Dec 14 2017 *)
f[n_] := With[{t = Round@ Sqrt@ n}, 1/2*(-1)^t*({1, -1}(Abs[t^2 - n] - t) + t^2 - n - Mod[t, 2])]; Table[f@ n, {n, 0, 95}] // Flatten (* Mikk Heidemaa May 23 2020, after Stephen Wolfram *)

apply( {coords(n)=my(m=sqrtint(n), k=m\/2); if(m <= n -= 4*k^2, [n-3*k,-k], n >= 0, [-k,k-n], n >= -m, [-k-n,k], [k,3*k+n])}, [0..99]) \\ Use concat(%) to remove brackets '[', ']'. This function gives the coordinates of n on the spiral starting with 0 at (0,0), as shown in Examples for A174344, A274923, ..., so (a(2n-1),a(2n)) = coords(n-1). To start with 1 at (0,0), change n to n-=1 in sqrtint(). The inverse function is pos(x,y) given e.g. in A316328. - M. F. Hasler, Oct 20 2019

from math import ceil, sqrt
def get_coordinate(n):
    k=ceil((sqrt(n)-1)/2)
    t=2*k+1
    m=t**2
    t=t-1
    if n >= m - t:
        return k - (m-n), -k
    else:
        m -= t
    if n >= m - t:
        return -k, -k+(m-n)
    else:
        m -= t
    if n >= m-t:
        return -k+(m-n), k
    else:
        return k, k-(m-n-t)

a(2*n-1) = A174344(n).
a(2*n) = A274923(n) = -A268038(n).
abs(a(n+2) - a(n)) < 2.
a(2*n-1)+a(2*n) = A180714(n).
f(n) = floor(-n/4)*ceiling(-3*n/4 - 1/4) mod 2 + ceiling(n/8) (gives the pairs of coordinates for integers in the diagonal rays). - Mikk Heidemaa, May 07 2020

A334742 Pascal's spiral: starting with a(1) = 1, proceed in a square spiral, computing each term as the sum of horizontally and vertically adjacent prior terms.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 10, 12, 12, 14, 17, 20, 20, 23, 27, 32, 37, 37, 42, 48, 55, 62, 62, 69, 77, 87, 99, 111, 111, 123, 137, 154, 174, 194, 194, 214, 237, 264, 296, 333, 370, 370, 407, 449, 497, 552, 614, 676, 676, 738, 807, 884, 971, 1070

Offset: 1

Alec Jones and Peter Kagey, May 09 2020

This is the square spiral analogy of Pascal's triangle thought of as a table read by antidiagonals.

			Spiral begins:
  111--99--87--77--69--62
                        |
   12--12--10---8---7  62
    |               |   |
   14   2---2---1   7  55
    |   |       |   |   |
   17   3   1---1   6  48
    |   |           |   |
   20   3---4---5---5  42
    |                   |
   20--23--27--32--37--37
a(15) = 10 = 8 + 2, the sum of the cells immediately to the right and below. The term to the left is not included in the sum because it has not yet occurred in the spiral.

Peter Kagey, Table of n, a(n) for n = 1..10000
Peter Kagey, Bitmap illustrating the parity of the first 2^20=1048576 terms. (Even and odd numbers are represented with black and white pixels respectively.)

Cf. A007318, A141481, A334688.
Cf. A180714, A214526, A274640, A278180, A304586, A307834, A334741.
x- and y-coordinates are given by A174344 and A274923, respectively.

a(A033638(n)) = a(A002620(n)) for n > 1.

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

A334741 Fill an infinite square array by following a spiral around the origin; in the central cell enter a(0)=1; thereafter, in the n-th cell, enter the sum of the entries of those earlier cells that are in the same row or column as that cell.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 11, 21, 40, 47, 93, 180, 203, 397, 796, 1576, 1675, 3305, 6636, 13192, 14004, 27607, 55029, 110192, 220024, 226740, 450123, 898661, 1798700, 3594248, 3704800, 7354303, 14681369, 29349536, 58710640, 117394896, 119196748, 237492079

Offset: 0

Alec Jones and Peter Kagey, May 09 2020

nonn

The spiral track being used here is the same as in A274640, except that the starting cell here is indexed 0 (as in A274641).

The central cell gets index 0 (and we fill it in with the value a(0)=1).

			Spiral begins:
     3----2----1
     |         |
     5    1----1   47
     |              |
     8---11---21---40
a(11) = 47 = 1 + 1 + 5 + 40, the sum of the cells in its row and column.

Peter Kagey, Table of n, a(n) for n = 0..2499

Cf. A280027.
Cf. A180714, A214526, A274640, A278180, A304586, A307834, A334742.
x- and y-coordinates are given by A174344 and A274923, respectively.

\\ here P(n) returns A174344 and A274923 as pair.
P(n)={my(m=sqrtint(n), k=ceil(m/2)); n -= 4*k^2; if(n<0, if(n<-m, [k, 3*k+n], [-k-n, k]), if(nAndrew Howroyd, May 09 2020

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A174344 List of x-coordinates of point moving in clockwise square spiral.

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A296030 Pairs of coordinates for successive integers in the square spiral (counterclockwise).

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A334742 Pascal's spiral: starting with a(1) = 1, proceed in a square spiral, computing each term as the sum of horizontally and vertically adjacent prior terms.

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

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PARI

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A334741 Fill an infinite square array by following a spiral around the origin; in the central cell enter a(0)=1; thereafter, in the n-th cell, enter the sum of the entries of those earlier cells that are in the same row or column as that cell.

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