A396656 - OEIS

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.

%I #5 Jun 15 2026 19:47:47

%S 0,3,6,7,10,13,16,25,26,32,34,41,47,52,53,68,74,75,77,83,85,89,90,92,

%T 96,97,102,103,113,115,120,131,133,134,142,147,152,154,155,158,163,

%U 168,170,171,173,174,175,179,180,184,195,198,199,201,204,208,209,213

%N 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.

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

%C 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.

%C 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).

%H Nick J. Nauta, <a href="/A396656/a396656.png">The spiral after the first 1000 rounds</a>

%H Nick J. Nauta, <a href="/A396656/a396656_1.png">The spiral after the first 10000 rounds</a>

%H Nick J. Nauta, <a href="/A396656/a396656_2.png">The spiral after the first 100000 rounds</a>

%H Nick J. Nauta, <a href="/A396656/a396656_3.png">The spiral after the first 1000000 rounds</a>

%e The spiral begins:

%e B---R---*---G---*

%e / \

%e B *---R---R---* G

%e / / \ \

%e * * G---B---R B *

%e / / / \ \ \

%e G B R R---B G B R

%e / / / / \ \ \ \

%e R * B G R---G B * *

%e \ \ \ \ / / /

%e R B G B---R---R B G

%e \ \ \ / /

%e * R R---B---G---* *

%e \ \ /

%e G *---R---*---*---B

%e \

%e G---G---*---B---B

%Y Cf. A395365, A396330, A396653.

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

%K nonn

%O 1,2

%A _Nick J. Nauta_, Jun 11 2026