A340945 List of Y-coordinates of point moving along one of the arms of a counterclockwise square spiral with four arms; A340944 gives X-coordinates.

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

Offset: 0

Author

Rémy Sigrist, Jan 30 2021

Keywords

• sign
• easy
• look

Examples

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

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10101
• Rémy Sigrist, PARI program for A340945
• Index entries for sequences related to coordinates of 2D curves

Crossrefs

Cf. A340172, A340944, A000194.

Programs

PARI

See Links section.

PARI

a(n) = my(r,s=sqrtint(n,&r)); if(r>s, s++;r-=2*s-1); imag((r-s*I)*I^s); \\ Kevin Ryde, May 28 2023

Formula

a(n) = 0 iff n = 4*x^2 + 4*x + 1 for some x.

a(n) = Im z where z = (r-s*i)*i^s with s = round(sqrt(n)) = A000194(n) and remainder r = n - s^2. - Kevin Ryde, May 28 2023