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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

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 a(n) is their -x(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 17 2019

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

This is a 3D generalization of the 2D square spiral and could be used to produce a 3D variant of Ulam's prime spiral.

See A343630 for an analog using the Euclidean or 2-norm instead of the sup- or oo-norm used here, so points are partitioned in spheres and circles instead of squares and cubes here.

The integer lattice points, Z^3, are listed in order of increasing sup norm R = max(|x|, |y|, |z|). Each "sphere" or shell of given radius R is filled starting at the North or South pole using concentric squares on the top and bottom face and squares of fixed size (2R+1) X (2R+1) at intermediate z-coordinates. Each square (circle for the sup-norm) is filled in the sense of increasing longitude, where the positive x axis corresponds to longitude 0, i.e., the points (r,0,z), (0,r,z), (-r,0,z) and (0,-r,z) are visited in this order. The z-values are alternatively increasing and decreasing (so over a period of two shells they follow the same rectangle-wave shape as the x-values do over the period of each square).

The sequence can be seen as a table with row length of 3, where each row corresponds to the (x,y,z)-coordinates of one point (then the three columns are A343641, A343642 and A343643), or as a table with row lengths 3*A010014, where A010014(r) is the number of points with sup-norm r.

There are (2n+1)^3 integer lattice points with sup norm <= n. Therefore, the point number n (where 0 is the origin) is in the shell r = round(n^(1/3)/2) = floor(...+1/2). Within shell r, which starts with the point number (2r-1)^3 (except for r=0), the first and last (2r+1)^2 points are on square spirals on the top and bottom faces, and the other points are on 2r-1 squares forming "belts" of 8r points each, on the side faces of the cube.

The terms corresponding to the corner points of the spiral with a(k-1) < a(k) > a(k+1), i.e., 2, 2, 2, 5, 8, 8, 8, 13, 18, 18, 18, ... are given by the sequence A001105(1) repeated 3 times, (A001105(1)+A001105(2))/2, A001105(2) repeated 3 times, (A001105(2)+A001105(3))/2, A001105(3) repeated 3 times, ... .

These numbers are the norms of the Gaussian integers discussed in A345436. - N. J. A. Sloane, Jun 25 2021

Equivalently the number immediately below n in a counterclockwise spiral of the positive integers with the first step to the right.

This is a permutation of the positive integers. A334752 is the inverse permutation.

Equivalently the number immediately above n in a counterclockwise square spiral of the positive integers with the first step to the right.

This is a permutation of the positive integers. A334751 is the inverse permutation.

A329116 gives x-coordinates for a point moving in counterclockwise triangular spiral.

Any grid point with relative coordinates (x,y) from the central grid point, which is numbered 1, and where the greatest common divisor (gcd) of |x| and |y| equals 1 will be visible from the central point. Grid points where gcd(|x|,|y|) > 1 will have another point directly between it and the central point and will thus not be visible. In an infinite 2D square lattice the ratio of visible grid points to all points is 6/Pi^2, approximately 0.608, the same as the probability of two random numbers being relative prime.

For a square spiral of size 10001 X 10001, slightly over 100 million numbers, a total of 60803664 numbers are visible, of which 2155170 are prime. The total number of primes in the same range is 5762536, giving a ratio of visible primes to all primes of about 0.374. This is significantly lower than the ratio for all numbers of 0.608, indicating a prime is more likely to be hidden from the origin than a random number.

Primes p such that A174344(p) and A268038(p) are coprime. - Robert Israel, Feb 16 2024

P(n) is a corner on a spiral like this:

* * * * * * * * * * * *

*

* * * * * * * * *

* * *

* * * * * * *

* * * * *

* * * * * *

* * * *

* * * * * * * *

* *

* * * * * * * * * *

If we interpret the pointer from P(0) to P(n) as a complex number z(n), the description of the spiral is short because a 90-degree left turn is a multiplication by i (imaginary unit) and the distance of P(n) from P(0) is abs(z(n))^2, see formula 1.