A396330 Consider the square spiral with its cells numbered starting at 0. Three players, Red, Green and Blue, take turns and place knights at the smallest unoccupied cell not attacked by an opposing knight, where Red is only opposed by Blue, Green by Red and Blue by Green. Sequence lists squares occupied by a Red knight.

0, 3, 6, 10, 12, 13, 17, 19, 23, 30, 34, 37, 41, 44, 53, 54, 59, 64, 68, 72, 79, 85, 87, 89, 98, 104, 106, 108, 110, 112, 116, 118, 120, 122, 123, 124, 126, 139, 141, 147, 149, 155, 157, 163, 164, 165, 166, 167, 170, 174, 176, 178, 187, 193, 197, 199, 201, 203

Offset

1,2

Comments

This is a three-player variation of the two-player game on a square spiral (A392177).

Also, a variation of the three-player game on a square spiral (A395645), except that here players can be protected from certain other players' attacks. We use the simplest version: a cyclic protection where Red is protected from Green (only opposed by Blue), Green is protected from Blue (only opposed by Red), and Blue is protected from Red (only opposed by Green). If a player of color i is protected from a player of color j, then i may place pieces in cells under attack by Player j, effectively ignoring the threats from their attacks. Pieces may never be placed in occupied cells and pieces are never removed.

As the number of terms grows, the occupied cells form undulating bands that radiate outward from the center, with the three players' pieces alternating between bands. These regular bands are interspersed with irregular regions where pieces from all three players intermix (see linked illustrations).

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Michael S. Branicky, Python program for OEIS A396330, A396331 and A396332

Nick J. Nauta, The spiral after the first 1000 rounds

Nick J. Nauta, The spiral after the first 10000 rounds

Nick J. Nauta, The spiral after the first 100000 rounds

Nick J. Nauta, The spiral after the first 1000000 rounds

Example

The spiral begins:
   G--G--R--B--*--B--G--*--*--B--B  G
   |                             |  |
   B  R--B--*--G--*--R--B--B--G  R  B
   |  |                       |  |  |
   *  G  B--G--R--*--B--G--R  B  B  *
   |  |  |                 |  |  |  |
   B  *  R  B--B--G--R--R  B  R  R  B
   |  |  |  |           |  |  |  |  |
   R  G  *  R  G--R--B  *  *  R  B  *
   |  |  |  |  |     |  |  |  |  |  |
   G  R  *  *  B  R--G  R  *  G  R  R
   |  |  |  |  |        |  |  |  |  |
   R  G  B  R  R--G--B--B  G  *  *  G
   |  |  |  |              |  |  |  |
   B  B  R  G--B--G--R--G--B  G  G  R
   |  |  |                    |  |  |
   R  G  B--*--R--*--G--G--B--G  B  R
   |  |                          |  |
   B  R--*--B--G--*--*--*--R--G--G  R
   |                                |
   R--B--R--B--G--*--R--G--R--G--R--G

Prog

(Python) # See Links section.

Crossrefs

Cf. A392177, A395645, A396331, A396332.

Sequence in context: A261662 A345915 A050107 * A120068 A015875 A301285

Adjacent sequences: A396327 A396328 A396329 * A396331 A396332 A396333

Keyword

nonn,look,changed

Author

Nick J. Nauta, May 22 2026

Status

approved