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Number of fully symmetric recurrent sandpiles on an n X n grid.
1

%I #12 Dec 21 2022 04:34:54

%S 1,4,2,32,12,832,232,69632,14416,18719744,2876576

%N Number of fully symmetric recurrent sandpiles on an n X n grid.

%F A256046(n) divides a(n), a(n) divides A348566(n, n).

%e a(0) = 1, the empty sandpile on an empty grid.

%e a(1) = 4, all 4 sandpiles on a 1 X 1 grid are fully symmetric.

%e a(2) = 2 fully symmetric recurrent sandpiles on an 2 X 2 grid are the all-2 and all-3 sandpiles.

%e Among A348566(4, 4) = 36 sandpiles with horizontal and vertical symmetry, only a(4) = 12 also have diagonal symmetries:

%e 3333 1331 1331 3223 3223 3333 2222 0330 2332 2332 0330 2222

%e 3333 3223 3333 2222 2332 3223 2222 3223 3223 3333 3333 2332

%e 3333 3223 3333 2222 2332 3223 2222 3223 3223 3333 3333 2332

%e 3333 1331 1331 3223 3223 3333 2222 0330 2332 2332 0330 2222

%Y Cf. A348566, A256046.

%K nonn,more

%O 0,2

%A _Andrey Zabolotskiy_, Oct 22 2021