A010751 Up once, down twice, up three times, down four times, ...

0, 1, 0, -1, 0, 1, 2, 1, 0, -1, -2, -1, 0, 1, 2, 3, 2, 1, 0, -1, -2, -3, -2, -1, 0, 1, 2, 3, 4, 3, 2, 1, 0, -1, -2, -3, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -5, -4

Offset: 0

Author

David Berends (dave(AT)pgt.com)

Keywords

• sign
• easy
• tabl

Comments

Also x-coordinates of a point moving in a spiral rotated by Pi/4, with y-coordinates given by A305258. - Hugo Pfoertner, May 29 2018

This sequence is also obtained by reading alternately in ascending or descending way the antidiagonals of the array defined as A(i, j) = floor((j - i + 1)/2) (see Example). - Stefano Spezia, Jan 02 2022

Examples

			From _Stefano Spezia_, Jan 02 2022: (Start)
The array A begins with:
   0   1   1   2   2   3   3 ...
   0   0   1   1   2   2   3 ...
  -1   0   0   1   1   2   2 ...
  -1  -1   0   0   1   1   2 ...
  -2  -1  -1   0   0   1   1 ...
  -2  -2  -1  -1   0   0   1 ...
  ...
(End)
		

Links

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

Crossrefs

Cf. A000384, A002024, A007590, A014105, A305258.

Programs

Mathematica

n=(the index); x = -1; y = 0; While[n != 0, While[y != x && n != 0, y--; n-- ]; While[y != -x && n != 0, n--; y++ ]; x-- ]; Print[ -y] (* provided by Gregory Puleo *)
n = (the index); a = Floor[(-1 - Sqrt[1 + 8* n])/4]; b = -Floor[(1 - Sqrt[1 + 8*n])/4]; a + 1 - Sign[a*(2*a + 1) - b*(2*b + 1)]*(n - 2*a^2 - 3*a - 1) (* Mark Spindler, Mar 25 2004 *)

PARI

step=-1;print1(x=0,", ");for(stride=1,12,step=-step;for(k=1,stride,print1(x+=step,", "))) \\ Hugo Pfoertner, Jun 02 2018

Python

from math import isqrt
def A010751(n): return n-(m**2>>1) if (m:=isqrt(n+1<<3)+1>>1)&1 else (m**2>>1)-n # Chai Wah Wu, Jun 08 2025

Formula

a(n) = x + 1 - (sign(x(2x+1) - y(2y+1)))*(n-2x^2-3x-1) where x = floor((-1-sqrt(1+8n))/4), y = -floor((1-sqrt(1+8n))/4), sign(x) = abs(x)/x when x is not 0 and sign(0) = 0, floor(x) is the greatest integer less than or equal to x, sqrt(x) is the principal square root of x and abs(x) is the absolute value (or magnitude) of x. - Mark Spindler, Mar 25 2004

From David A. Corneth, Jun 02 2018: (Start)

a(A007590(k)) = a(floor(k^2 / 2)) = 0.

a(A000384(k)) = a(binomial(2 * k, 2)) = k, a new maximum so far.

a(A014105(k)) = a(binomial(2 * k + 1, 2)) = -k, a new minimum so far.

(End)

a(n) = (-1)^A002024(n+1)*(A007590(A002024(n+1))-n). - William McCarty, Jul 30 2021