A296030 - OEIS

%I #143 Jan 27 2025 21:11:36

%S 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,

%T -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,

%U 1,3,2,3,3,2,3,1,3,0,3,-1,3,-2,3,-3,3,-3,2

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

%C The spiral is also called the Ulam spiral, cf. A174344, A274923 (x and y coordinates). - _M. F. Hasler_, Oct 20 2019

%C 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

%C From _Robert G. Wilson v_, Dec 05 2017: (Start)

%C 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.

%C 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;

%C 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;

%C 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;

%C 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;

%C The union of the four sequences above is A033638.

%C (End)

%C 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

%C 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

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

%H Benjamin Mintz, <a href="/A296030/b296030.txt">Table of n, a(n) for n = 1..100000</a>

%H BackIssues.com, <a href="http://backissues.com/issue/Scientific-American-March-1964">Scientific American March 1964 back issue</a>

%H Scientific American, <a href="/A168022/a168022.pdf">March 1964 cover</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ulam_spiral">Ulam Spiral</a>.

%F a(2*n-1) = A174344(n).

%F a(2*n) = A274923(n) = -A268038(n).

%F abs(a(n+2) - a(n)) < 2.

%F a(2*n-1)+a(2*n) = A180714(n).

%F 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

%e The integer 1 occupies the initial position, so its coordinates are {0,0}; therefore a(1)=0 and a(2)=0.

%e The integer 2 occupies the position immediately to the right of 1, so its coordinates are {1,0}.

%e The integer 3 occupies the position immediately above 2, so its coordinates are {1,1}; etc.

%t 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 *)

%t 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 *)

%t 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 *)

%o (Python)

%o from math import ceil, sqrt

%o def get_coordinate(n):

%o     k=ceil((sqrt(n)-1)/2)

%o     t=2*k+1

%o     m=t**2

%o     t=t-1

%o     if n >= m - t:

%o         return k - (m-n), -k

%o     else:

%o         m -= t

%o     if n >= m - t:

%o         return -k, -k+(m-n)

%o     else:

%o         m -= t

%o     if n >= m-t:

%o         return -k+(m-n), k

%o     else:

%o         return k, k-(m-n-t)

%o (PARI) 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

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

%Y 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).

%K sign,easy,look,changed

%O 1,19

%A _Benjamin Mintz_, Dec 03 2017