

A180238


a(n) is the number of distinct billiard words with length n on an alphabet of 3 symbols.


4



1, 3, 9, 27, 75, 189, 447, 951, 1911, 3621, 6513, 11103, 18267, 29013, 44691, 67251, 98547, 140865, 197679, 272799, 370659, 497403, 658371, 859863, 1110453, 1420527, 1799373, 2260161, 2815401, 3479235, 4269279
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OFFSET

0,2


COMMENTS



LINKS

L. Vuillon, Balanced Words, Bull. Belg. Math. Soc., 10 (2003), 787805.


FORMULA

Computation may be expedited by generating only words in which the symbols occur in increasing alphabetic order: this was done in the production version.


EXAMPLE

For n = 5 there are a(5) = 189 words, permutations on the alphabet {1,2,3} of the 32 words
11111, 11112, 11121, 11123, 11211, 11212, 11213, 11231, 12111, 12112, 12113, 12121, 12122, 12123, 12131, 12132, 12212, 12213, 12221, 12222, 12223, 12231, 12232, 12311, 12312, 12313, 12321, 12322, 12323, 12331, 12332, 12333.


MATHEMATICA

(* Number of ways to interleave N elements from 3 arithmetic seqs *)
(* Given a string like "ABCABA", produce a set of inequalities *)
(* about the three arithmetic progressions giving successive A/B/Cs *)
(* The Nth occurrence (1indexed) of character X corresponds to the value *)
(* BASE[X] + N * DELTA[X] *)
(* In all functions, seq is eg {"A", "B", "C", "A", "B", "A"} *)
(* The arithmeticprogression value of the ith element of seq *)
value[seq_, i_] := BASE[seq[[i]]] + DELTA[seq[[i]]] * numoccur[seq, i]
numoccur[seq_, i_] := Count[Take[seq, If[i>0, i, Length[seq]+i+1]], seq[[i]]]
(* First element of the seq is greater than anything that would precede it*)
lowerbound[seq_] := (BASE[ # ] < value[seq, 1])& /@ Union[seq]
(* Each element of the seq is greater than the previous one *)
upperbound[seq_] := (value[seq, 1] < value[Append[seq, # ], 1])& /@ Union[seq]
(* Last element of the seq is less than anything that would follow it *)
ordering[seq_] := Table[value[seq, i] < value[seq, i+1], {i, Length[seq]1}]
ineqs[seq_] := Join[ lowerbound[seq], ordering[seq], upperbound[seq] ]
vars[seq_] := Join @@ ({BASE[ # ], DELTA[ # ]}& /@ Union[seq])
witness[seq_] := FindInstance[ ineqs[seq], vars[seq] ]
witness[s_String] := witness[Characters[s]]
(* All obtainable lengthn shuffles of three arithmetic seqs: *)
names = {"A", "B", "C"}
shuf[0] := {""}
candidates[n_] := Flatten[Table[ob<>ch, {ob, shuf[n1]}, {ch, names}]]
shuf[n_] := shuf[n] = Select[ candidates[n], witness[ # ] != {}& ]
(* Typical session *)
In[18]:= Table[Length[shuf[i]], {i, 0, 12}]
Out[18]= {1, 3, 9, 27, 75, 189, 447, 951, 1911, 3621, 6513, 11103, 18267}
In[19]:= TimeUsed[]/60 Out[19]= 6.73642


CROSSREFS



KEYWORD

nonn,more


AUTHOR



STATUS

approved



