A396656 Consider the 0-indexed spiral on a hexagonal grid, as in A395365. 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 cells occupied by Red knights.

0, 3, 6, 7, 10, 13, 16, 25, 26, 32, 34, 41, 47, 52, 53, 68, 74, 75, 77, 83, 85, 89, 90, 92, 96, 97, 102, 103, 113, 115, 120, 131, 133, 134, 142, 147, 152, 154, 155, 158, 163, 168, 170, 171, 173, 174, 175, 179, 180, 184, 195, 198, 199, 201, 204, 208, 209, 213

Offset

1,2

Comments

These are Glinski knights, as in hexagonal chess (see A395362).

Hexagonal variation of the game on a square spiral (A396330), and a variation of A396653 in which players can be protected from certain other players' attacks.

As the number of terms grows, occupied cells organize into undulating concentric bands that radiate outward from the center, with the three players' colors alternating cyclically between bands (similar to A396330). Most bands form closed loops, but some adjacent bands merge or split near the radial axes of the hexagon, producing localized defects (see linked images).

Links

Table of n, a(n) for n=1..58.

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:
         B---R---*---G---*
        /                 \
       B   *---R---R---*   G
      /   /             \   \
     *   *   G---B---R   B   *
    /   /   /         \   \   \
   G   B   R   R---B   G   B   R
  /   /   /   /     \   \   \   \
 R   *   B   G   R---G   B   *   *
  \   \   \   \         /   /   /
   R   B   G   B---R---R   B   G
    \   \   \             /   /
     *   R   R---B---G---*   *
      \   \                 /
       G   *---R---*---*---B
        \
         G---G---*---B---B

Crossrefs

Cf. A395365, A396330, A396653.

See A396657 for the Green knight, A396658 for the Blue knight.

Sequence in context: A288215 A284625 A047281 * A182909 A269903 A191103

Adjacent sequences: A396653 A396654 A396655 * A396657 A396658 A396659

Keyword

nonn

Author

Nick J. Nauta, Jun 11 2026

Status

approved