A396116 Consider the square spiral with its cells numbered starting at 0. Four players take turns and place knights at the smallest unoccupied cell not attacked by an existing knight of a different player. Sequence lists squares occupied by the first player.

0, 11, 17, 30, 36, 47, 50, 58, 59, 61, 86, 89, 92, 101, 116, 128, 132, 138, 144, 162, 168, 180, 186, 187, 191, 193, 214, 239, 240, 242, 244, 248, 257, 276, 294, 298, 300, 302, 306, 312, 318, 324, 348, 360, 378, 381, 384, 385, 386, 391, 395, 397, 404, 420, 456

Offset

1,2

Comments

A variant of A392177 with four players.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Brady Haran, Jonas Karlsson, and N. J. A. Sloane, Red & Black Knights (extraordinary result); Amazing Chessboard Patterns (extra), YouTube Numberphile videos, May 2026

Jonas Karlsson, Four knights with colors black, red, cyan, purple. Close-up, 200 of each color.

Jonas Karlsson, Four knights with colors black, red, cyan, purple. Zoom-out, 60K of each color.

Scott R. Shannon, Four knights with colors Red, Green, Blue, Yellow, after ten million knights have been placed.

Rémy Sigrist, PARI program

Example

The spiral begins:
     *---*---*---*---*---*---*---*--#1---*--#3
     |                                       |
    #1  #2--#4--#3--#1--#2--#1--#1--#3--#3  #1
     |   |                               |   |
    #4  #3  #1---*---*---*---*---*--#1   *  #3
     |   |   |                       |   |   |
     *   *   *   *---*---*---*---*   *   *   *
     |   |   |   |               |   |   |   |
    #3   *  #3  #1  #2--#4--#3  #1  #2   *  #1
     |   |   |   |   |       |   |   |   |   |
    #3   *   *  #4  #3  #1--#2  #4   *   *   *
     |   |   |   |   |           |   |   |   |
    #4   *   *   *   *---*---*---*   *   *   *
     |   |   |   |                   |   |   |
    #2  #3  #4  #3---*---*---*--#2--#4  #1  #2
     |   |   |                           |   |
    #4   *  #2--#3--#4---*--#4--#1--#2--#2  #4
     |   |                                   |
     *   *---*--#4---*---*---*---*---*---*---*
     |
    #2---*--#3---*--#3--#4--#1---*---*---*--#2

Prog

(PARI) \\ See Links section.

Crossrefs

Cf. A392177, A396117, A396118, A396119, A396120, A396121.

Sequence in context: A211486 A208573 A142718 * A258866 A189324 A127703

Adjacent sequences: A396111 A396114 A396115 * A396117 A396118 A396119

Keyword

nonn

Author

Rémy Sigrist, May 20 2026

Status

approved