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A072844 Number of words of length 2n-1 generated by the two letters s and t that reduce to the identity 1 by 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.

%I A072844 #17 Nov 11 2024 11:18:55
%S A072844 0,0,0,1,9,55,286,1365,6188,27132,116281,490337,2043275,8439210,
%T A072844 34621041,141290436,574274008,2326683921,9402807817,37923176863,
%U A072844 152705590518,614111175965,2467123420524,9903167265124,39725253489545
%N A072844 Number of words of length 2n-1 generated by the two letters s and t that reduce to the identity 1 by 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.
%D A072844 H.S.M. Coxeter and W.O.J. Moser, Generators and Relations for Discrete Groups, Fourth Edition, (p.134).
%H A072844 Haggai Liu, <a href="http://www.koutschan.de/conf/ACA21/ACA2021_slides_Liu.pdf">Enumerative Properties of Cogrowth Series on Free Products of Finite Groups</a>, ACA 2021 Session on Algorithmic Combinatorics, 2021.
%F A072844 a(n) = 9*a(n-1) - 26*a(n-2) + 25*a(n-3) - 4*a(n-4).
%F A072844 g.f.: x^4 / ((1 - 4*x)*(1 - 5*x + 6*x^2 - x^3)). - _Colin Barker_, Feb 24 2017
%F A072844 28*a(n) = 4^n -4*( 2*A005021(n) -9*A005021(n-1) +11*A005021(n-2) ). - _R. J. Mathar_, Nov 05 2024
%e A072844 The 9 words of length 9 are ssssssstt, sssssstts, sssssttss, ssssttsss, sssttssss, ssttsssss, sttssssss, ttsssssss, tssssssst. - _Sean A. Irvine_, Oct 31 2024
%Y A072844 Cf. A072266.
%Y A072844 Bisection of A377573.
%K A072844 nonn
%O A072844 1,5
%A A072844 Jamaine Paddyfoot and _John W. Layman_, Jul 24 2002