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A265434 Consider the group freely generated by an element U of order 3 and an element S of order 2. a(n) gives the number of words in the alphabet {U,S} of length n that are equal to unity. 3
1, 0, 1, 1, 1, 5, 2, 14, 13, 31, 66, 77, 240, 286, 722, 1226, 2141, 4760, 7268, 16473, 27716, 54615, 106217, 187818, 388084, 685830, 1370162, 2569351, 4849538, 9526355, 17598392, 34694301, 65152925, 125679731, 242019753, 459049449, 893614680, 1695840536 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

One of the models of such a group is the full modular group PSL_2(Z).

LINKS

Robert Israel, Table of n, a(n) for n = 0..2000

G. Alkauskas, The modular group and words in its two generators, arXiv:1512.02596 [math.NT] 2015-2017; Lithuanian Math. J. 57 (1) (2017), 1-12.

Jason Bell, Marni Mishna, On the Complexity of the Cogrowth Sequence, arXiv:1805.08118 [math.CO], 2018.

FORMULA

G.f.: G(x) satisfies (6*x^5-3*x^4+2*x^3+3*x^2-1)*G(x)^3 + (x^5-x^4+x^3+2*x^2-1)*G(x)^2 + (x^3-x^2+1)*G(x) + 1 = 0.

EXAMPLE

For n=5 the a(n)=5 words are U^2S^2U, US^2U^2, S^2U^3, SU^3S, U^3S^2.

MAPLE

simpl:= proc(W) local A, Ap;

   Ap:= W;

   while A <> Ap do

     A:= Ap;

     Ap:= StringTools:-RegSubs("UUU"="", StringTools:-RegSubs("SS"="", A));

   od;

   A;

end proc:

F:= proc(n, W) option remember;

  if n = 0 then

     if W = "" then return 1 else return 0 fi;

  fi;

  procname(n-1, simpl(cat("S", W))) + procname(n-1, simpl(cat("UU", W)))

end proc:

seq(F(n, ""), n=0..40); # Robert Israel, Dec 09 2015

MATHEMATICA

(* Program not suitable to compute a large number of terms *)

a[n_] := Count[Tuples[{U, S}, n] //. {A___, U, U, U, B___} | {A___, S, S, B___} -> {A, B}, {}];

Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 20}] (* Jean-Fran├žois Alcover, Jan 18 2018 *)

CROSSREFS

Cf. A276408 (primitive words).

Sequence in context: A060422 A213751 A185781 * A189236 A191722 A191435

Adjacent sequences:  A265431 A265432 A265433 * A265435 A265436 A265437

KEYWORD

nonn

AUTHOR

Giedrius Alkauskas, Dec 09 2015

STATUS

approved

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Last modified December 17 06:45 EST 2018. Contains 318192 sequences. (Running on oeis4.)