A329116 Successively count to (-1)^(n+1)*n (n = 0, 1, 2, ... ).

0, 1, 0, -1, -2, -1, 0, 1, 2, 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, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8

Offset: 0

Author

Mikk Heidemaa, Nov 13 2019

Keywords

• sign
• easy
• look

Comments

Also x-coordinates of a point moving in counterclockwise triangular spiral (A329972 gives the y-coordinates).

Examples

			   y
     |
   4 |                         56
     |                           \
     |                            \
     |                             \
   3 |                         30  55
     |                         / \   \
     |                        /   \   \
     |                       /     \   \
   2 |                     31  12  29  54
     |                     /   / \   \   \
     |                    /   /   \   \   \
     |                   /   /     \   \   \
   1 |                 32  13   2  11  28  53
     |                 /   /   / \   \   \   \
     |                /   /   /   \   \   \   \
     |               /   /   /     \   \   \   \
   0 |             33  14   3   0---1  10  27  52
     |             /   /   /             \   \   \
     |            /   /   /               \   \   \
     |           /   /   /                 \   \   \
  -1 |         34  15   4---5---6---7---8---9  26  51
     |         /   /                             \   \
     |        /   /                               \   \
     |       /   /                                 \   \
  -2 |     35  16--17--18--19--20--21--22--23--24--25  50
     |     /                                             \
     |    /                                               \
     |   /                                                 \
  -3 | 36--37--38--39--40--41--42--43--44--45--46--47--48--49
     |
     +--------------------------------------------------------
   x:  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7
We count as follows. Start at n=0 with 0.
Next step is to count to 1: so we have 0, 1.
Next step is to count to -2, so we have 0, 1, 0, -1, -2.
Next we have to go to +3, so we have 0, 1, 0, -1, -2, -1, 0, 1, 2, 3.
And so on.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to coordinates of 2D curves

Crossrefs

Cf. A053615, A196199, A339265 (first differences). Essentially the same as A255175.

Programs

Mathematica

a[n_] := Table[(-1)^(# + 1)*(-#^2 + # + k) &[Ceiling@ Sqrt@ k], {k, 0, n}]; a[64]

Python

from math import isqrt
def A329116(n): return ((t:=1+isqrt(n-1))*(t-1)-n)*(-1 if t&1 else 1) if n else 0 # Chai Wah Wu, Aug 04 2022

Formula

a(n) = (-1)^t * (t^2 - t - n) where t=ceiling(sqrt(n)).

a(n) = (-1)^t * floor(t^2 - sqrt(n) - n) where t=ceiling(sqrt(n)).

A053615(n) = abs(a(n)).

abs(A196199(n)) = abs(a(n)).

A255175(n) = a(n+1).