cp's OEIS Frontend

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

From Wolfdieter Lang, Oct 24 2020: (Start)

If this sequence is written with offset 1 as a number triangle T(n, k), with n the length of row n, for n >= 1, then row n gives the primitive period of the periodic sequence {k (mod* n)}_{k>=0}, where k (mod* n) = k (mod n) if k <= floor(n/2) and otherwise it is -k (mod n). Such a modified modular relation mod* n has been used by Brändli and Beyne, but for integers relative prime to n.

These periodic sequences are given in A000007, A000035, A011655, A007877, |A117444|, A260686, A279316, for n = 1, 2, ..., 7. For n = 10 A271751, n = 12 A271832, n = 14 A279313. (End)

This sequence uses a diagonal spiral of numbers to enumerate the squares on the board. The knight starts on the square with number 1. At each step the knight goes to an unvisited square with the smallest number.

The sequence if finite. After 3722 steps the square with number 3541 is visited, after which all neighboring squares have been visited.

The infinite 2D grid is labeled along a diamond spiral as shown in A305258, starting with 0 at the origin (0,0), where each "shell" contains the points with given taxicab or L1-norm, as follows:

. (y)

2 | 8 17

| / \ \

1 | 9 2 7 16

| / / \ \ \

0 | 10 3 0--1 6 15

| \ \ / /

-1 | 11 4--5 14

| \ /

-2 | 12--13

|___________________

x: -2 -1 0 1 2 3

.

(This numbering, where the n-th "shell" has only 4n numbers, is "finer" than the square spiral numbering where the n-th shell has 8n numbers.)

The cursor is moving like a chess king to the von Neumann neighbor not visited earlier and labeled with the smallest prime number if possible, otherwise with the smallest possible composite number.

After the 92th move, the cursor is trapped in the point (-1,-3) labeled a(92) = 39. All eight neighbors were then already visited earlier, so the king has no more any possible move: see the "path plot" given in the links section.

A permutation of the nonnegative integers, whose restriction to the positive integers is also a permutation. See A383189 for the inverse permutation.

By "king" we simply mean a cursor that can move to any of the 8 nearest (von Neumann) neighbors, as the king moves in the game of chess.

Analog of A383185 (which uses the square spiral or shells of given sup norm) for the diamond spiral (or shells of given taxicab or L1-norm) as defined, e.g., in A305258. There are less grid points having a given L1 norm, so, in this sense, the diamond spiral numbering yields a "finer" measure of the distance from the origin than the square spiral numbering.

Instead of introducing a wall that has to be crossed (as in A383185), it is here sufficient to forbid the immediately following or preceding point in order to avoid a trivial sequence (0, 1, 2, ...) of numbers.

The actual trajectory of the king is however relatively uninteresting: The resulting path consists in a clockwise square spiral (centered at (1/2, 1/2)) with transition to the next shell on the negative y-axis, cf. the "path plot" given in links.