A396129 Consider the square spiral with its cells numbered starting at 0. Six 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, 21, 30, 40, 42, 53, 89, 91, 111, 132, 144, 152, 154, 156, 179, 185, 204, 212, 239, 241, 246, 266, 276, 299, 306, 311, 326, 334, 344, 357, 371, 377, 383, 402, 414, 424, 451, 461, 463, 482, 492, 500, 525, 531, 541, 552, 568, 574, 582, 592, 604, 615, 625

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.

Scott R. Shannon, Six knights after 2500000 knights have been placed.

Rémy Sigrist, PARI program

Example

The spiral begins:
    #3---*--#2---*--#3--#4--#4---*---*--#1---*
     |                                       |
     *  #6---*--#4---*--#6---*---*--#4--#3  #1
     |   |                               |   |
    #5   *  #5---*--#4---*---*---*--#1  #2   *
     |   |   |                       |   |   |
     *  #3   *  #4---*---*---*---*  #3   *   *
     |   |   |   |               |   |   |   |
    #2   *  #6   *  #2--#4--#3  #1  #2  #1   *
     |   |   |   |   |       |   |   |   |   |
     *  #5   *   *   *  #1--#2   *  #3   *  #2
     |   |   |   |   |           |   |   |   |
    #5   *  #1   *   *---*---*---*   *   *   *
     |   |   |   |                   |   |   |
     *  #4  #5  #5--#1--#2---*---*--#6   *  #6
     |   |   |                           |   |
     *   *  #1---*--#5--#5---*--#2--#6--#3  #6
     |   |                                   |
     *  #5---*---*---*---*---*---*--#6---*--#6
     |
    #3--#1--#4---*--#5--#5---*--#6---*--#6---*

Prog

(PARI) \\ See Links section.

Crossrefs

Cf. A392177, A396130, A396131, A396132, A396133, A396134, A396135, A396166.

Sequence in context: A195100 A125164 A101223 * A109686 A077522 A308232

Adjacent sequences: A396126 A396127 A396128 * A396130 A396131 A396132

Keyword

nonn

Author

Rémy Sigrist, May 22 2026

Status

approved