A320731 - OEIS

%I #11 Nov 16 2024 17:30:28

%S 1,3,9,24,60,141,328,738,1647,3618,7893,17055,36619,78144,165888,

%T 350619,738012,1548279,3237611,6752439,14046525,29157612,60396996,

%U 124885167

%N Number of possible states when placing n tokens of 2 alternating types on 3 piles.

%C Piles start empty and have no height limit. A token can only be placed on top of a pile. The starting token is fixed.

%e With alternating symbols A and B on three piles (starting with A), the following states emerge after placing 3 symbols in all 3^3 possible ways:

%e    1    2    3    4    5    6    7    8    9    10   11   12   13   14   15

%e   A                                                                 A

%e   B    B    B    A     A        A           A  A     A         B    B    B

%e   A__  AA_  A_A  AB_  AB_  ABA  A_B  AAB  A_B  BA_  BA_  BAA  AA_  _A_  _AA

%e    16   17   18   19   20   21   22   23   24   25   26   27

%e                                                            A

%e         A     A  A           A        A     A    B    B    B

%e   AAB  _AB  _AB  B_A  BAA  B_A  ABA  _BA  _BA  A_A  _AA  __A

%e 3 pairs of states (numbered (6,22), (8,16) and (12,20)) are identical, all others are different, hence a(3)=24.

%o (Python)

%o def fill(patterns, state_in, ply_nr, n_plies, n_players, n_stacks):

%o     if ply_nr>=n_plies:

%o         patterns.add(tuple(state_in))

%o     else:

%o         symbol=chr(ord('A')+ply_nr%n_players)

%o         for st in range(n_stacks):

%o             state_out=list(state_in)

%o             state_out[st]+=symbol

%o             fill(patterns, state_out, ply_nr+1, n_plies, n_players, n_stacks)

%o def A320731(n):

%o     n_plies, n_players, n_stacks = n, 2, 3

%o     patterns=set()

%o     state=[""]*n_stacks

%o     fill(patterns, state, 0, n_plies, n_players, n_stacks)

%o     return len(patterns)

%Y For 2 token types on 2 piles, see A320452.

%K nonn,more

%O 0,2

%A _Bert Dobbelaere_, Oct 20 2018