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Number of words of length 2n generated by the two letters s and t that reduce to the identity 1 using the relations sssssss=1, tt=1 and stst=1. The generators s and t along with the three relations generate the 14-element dihedral group D7.
3

%I #16 Nov 05 2024 07:15:48

%S 1,1,3,10,35,126,462,1717,6451,24463,93518,360031,1394582,5430530,

%T 21242341,83411715,328589491,1297937234,5138431851,20380608990,

%U 80960325670,322016144629,1282138331587,5109310929719,20374764059254

%N Number of words of length 2n generated by the two letters s and t that reduce to the identity 1 using the relations sssssss=1, tt=1 and stst=1. The generators s and t along with the three relations generate the 14-element dihedral group D7.

%H Colin Barker, <a href="/A072266/b072266.txt">Table of n, a(n) for n = 0..1000</a>

%H H. S. M. Coxeter and W. O. J. Moser, <a href="http://doi.org/10.1007/978-3-662-21943-0">Generators and Relations for Discrete Groups</a>, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 134.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (9,-26,25,-4).

%F G.f.: 1 -x*(2*x-1)*(x^2-4*x+1)/((4*x-1)*(x^3-6*x^2+5*x-1)). - _Michael Somos_, Jul 21, 2002

%F a(n) = 9*a(n-1) - 26*a(n-2) + 25*a(n-3) - 4*a(n-4) for n>4. - _Colin Barker_, Apr 26 2019

%F 14*a(n) = 4^n +2*(3*A005021(n) -10*A005021(n-1) +6*A005021(n-2)), n>0. - _R. J. Mathar_, Nov 05 2024

%e The words tttt=tsts=stst=1 so a(2)=3.

%t LinearRecurrence[{9,-26,25,-4},{1,1,3,10,35},30] (* _Harvey P. Dale_, Apr 16 2022 *)

%o (PARI) a(n)=if(n<1,n==0,sum(k=-(n-1)\7,(n-1)\7,C(2*n-1,n+7*k)))

%o (PARI) Vec((1 - 8*x + 20*x^2 - 16*x^3 + 2*x^4) / ((1 - 4*x)*(1 - 5*x + 6*x^2 - x^3)) + O(x^30)) \\ _Colin Barker_, Apr 26 2019

%Y Bisection of A377573.

%K nonn,easy,changed

%O 0,3

%A Jamaine Paddyfoot (jay_paddyfoot(AT)hotmail.com) and _John W. Layman_, Jul 08 2002