A396167 Consider the square spiral with its cells numbered starting at 0. Seven 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, 17, 23, 36, 40, 83, 110, 119, 144, 148, 156, 169, 181, 194, 206, 212, 229, 255, 270, 273, 289, 311, 324, 326, 340, 342, 346, 394, 396, 414, 422, 424, 455, 479, 483, 502, 514, 531, 545, 563, 572, 576, 598, 613, 627, 641, 666, 668, 670, 700, 712, 714, 737

Offset

1,2

Comments

A variant of A392177 with seven players.

Links

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

Azeez Daoud, Spiral Knight, Website for displaying up to 8 knights with different step sizes and colors on the square spiral board.

Azeez Daoud, Source code for above program

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, Seven knights with colors red, green, blue, yellow, purple, orange, grey, after ten million knights have been placed.

Scott R. Shannon, Seven knights with colors red, green, blue, yellow, purple, orange, grey, after 75 million knights have been placed.

Rémy Sigrist, PARI program

Example

The spiral begins:
    #4---*--#5---*---*--#2---*---*---*--#7---*
     |                                       |
     *  #2--#4--#2---*--#3--#7---*--#5--#3  #7
     |   |                               |   |
     *   *  #1---*---*---*---*--#3--#7  #4   *
     |   |   |                       |   |   |
     *   *  #2  #4---*---*---*---*  #3   *   *
     |   |   |   |               |   |   |   |
    #2   *  #5  #1  #2--#4--#3   *   *  #7   *
     |   |   |   |   |       |   |   |   |   |
     *   *   *   *   *  #1--#2   *  #3   *  #4
     |   |   |   |   |           |   |   |   |
    #6   *  #1   *   *---*---*---*   *  #7   *
     |   |   |   |                   |   |   |
     *  #6  #6  #5---*--#2--#1---*--#6   *  #1
     |   |   |                           |   |
    #6   *  #7---*---*--#5--#6--#2---*--#4   *
     |   |                                   |
     *  #6---*---*---*---*---*---*--#6---*--#5
     |
    #1---*--#3---*--#5--#6---*--#6---*--#1---*

Prog

(PARI) \\ See Links section.

Crossrefs

Cf. A392177, A396168, A396169, A396170, A396171, A396172, A396173, A396174, A396175.

Sequence in context: A080830 A165566 A072184 * A377172 A107644 A158710

Adjacent sequences: A396164 A396165 A396166 * A396168 A396169 A396170

Keyword

nonn

Author

Rémy Sigrist, May 23 2026.

Status

approved