A340172 List of Y-coordinates of point moving along one of the arms of a counterclockwise double square spiral; A340171 gives X-coordinates.

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

Offset: 0

Author

Rémy Sigrist, Dec 30 2020

Keywords

• sign

Comments

The odd function f such that f(n) = (A340171(n), a(n)) for any n >= 0 will visit exactly once every lattice point (so it is a bijection from Z to Z^2).

Examples

			The spiral starts as follows:
      +-----+-----+-----+-----+-----+
      .                             |
      .                             |
      .     +-----+-----+-----+     +
      .     |5     4     3    |2    |
      .     |                 |     |
            +     +-----+-----+     +
            |6    |      0     1    |     .
            |     |                 |     .
            +     +-----+-----+-----+     .
            |7                            .
            |                             .
            +-----+-----+-----+-----+-----+
             8     9     10    11    12    13
- so a(0) = a(1) = a(6) = 0,
-    a(2) = a(3) = a(4) = a(5) = 1.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Rémy Sigrist, PARI program for A340172
• Index entries for sequences related to coordinates of 2D curves

Crossrefs

Cf. A000217, A000384, A001105, A046092, A139274, A139275, A274923, A340171.

Programs

PARI

See Links section.

Formula

abs(a(n+1)-a(n)) + abs(A340171(n+1)-A340171(n)) = 1.

a(n) = A340171(n) iff n belongs to A001105.

a(n) = - A340171(n) iff n belongs to A046092.

2*a(n) = A340171(n) iff n belongs to A139274.

a(n) = 2*A340171(n) iff n belongs to A139275.

a(n) * A340171(n) = 0 iff n belongs to A000217.

a(n) = 0 iff n belongs to A000384.