A265434 - OEIS

%I #51 Oct 24 2023 08:52:58

%S 1,0,1,1,1,5,2,14,13,31,66,77,240,286,722,1226,2141,4760,7268,16473,

%T 27716,54615,106217,187818,388084,685830,1370162,2569351,4849538,

%U 9526355,17598392,34694301,65152925,125679731,242019753,459049449,893614680,1695840536

%N 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.

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

%H Robert Israel, <a href="/A265434/b265434.txt">Table of n, a(n) for n = 0..2000</a>

%H G. Alkauskas, <a href="http://arxiv.org/abs/1512.02596">The modular group and words in its two generators</a>, arXiv:1512.02596 [math.NT], 2015-2017.

%H G. Alkauskas, <a href="https://doi.org/10.1007/s10986-017-9339-2">The modular group and words in its two generators</a>, Lithuanian Math. J. 57(1) (2017), 1-12.

%H Jason Bell and Marni Mishna, <a href="https://arxiv.org/abs/1805.08118">On the Complexity of the Cogrowth Sequence</a>, arXiv:1805.08118 [math.CO], 2018.

%F 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.

%F a(n) ~ sqrt(s*(5*r^3*s*(6*s+1) - 4*r^2*s*(3*s+1) + 3*r*(2*s^2+s+1) + 6*s^2+4*s-2) / (r^5*(18*s+1) - r^4*(9*s+1) + r^3*(6*s+1) + r^2*(9*s+2) - 3*s - 1)) / (2*sqrt(Pi)*n^(3/2)*r^(n-1)), where r = 0.5072330945238052458926282360080236... and s = 5.833428896122262867435103255185854... are real roots of the system of equations (r^3-r^2+1)*s + (6*r^5-3*r^4+2*r^3+3*r^2-1)*s^3 + (r^5-r^4+r^3+2*r^2-1)*s^2+1 = 0, r^3 + 3*(6*r^5-3*r^4+2*r^3+3*r^2-1)*s^2 + 2*(r^5-r^4+r^3+2*r^2-1)*s+1 = r^2. - _Vaclav Kotesovec_, Oct 24 2023

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

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

%p    Ap:= W;

%p    while A <> Ap do

%p      A:= Ap;

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

%p    od;

%p    A;

%p end proc:

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

%p   if n = 0 then

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

%p   fi;

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

%p end proc:

%p seq(F(n,""), n=0..40); # _Robert Israel_, Dec 09 2015

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

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

%t Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 20}] (* _Jean-François Alcover_, Jan 18 2018 *)

%Y Cf. A276408 (primitive words), A276409 (reduced words).

%K nonn

%O 0,6

%A _Giedrius Alkauskas_, Dec 09 2015