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%I #21 Feb 18 2023 21:05:41
%S 0,6,12,72,240,1056,4032,16512,65280,262656,1047552,4196352,16773120,
%T 67117056,268419072,1073774592,4294901760,17180000256,68719214592,
%U 274878431232,1099510579200,4398048608256,17592181850112
%N Number of words of length 2n in the two letters s and t that reduce to the identity 1 by using the relations ssTT=1, ststSS=1 and ststTT=1, where S and T are the inverses of s and t, respectively (i.e., sS=1 and tT=1). The generators s and t and the three stated relations generate the quaternion group Q4.
%C a(n) = A003683(n+1)/6. No words of odd length (see the description above) reduce to 1.
%H G. C. Greubel, <a href="/A071930/b071930.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,8).
%F a(n) = 2^(2*n-2) - (-2)^(n-1) = 6*A003683(n-1).
%F From _Harvey P. Dale_, Dec 10 2012: (Start)
%F a(n) = 2*a(n-1) + 8*a-(n-2).
%F G.f.: 6*x/(1-2*x-8*x^2)). (End)
%F G.f.: Q(0), where Q(k)= 1 - 1/(4^k - 4*x*16^k/(4*x*4^k - 1/(1 + 1/(2*4^k - 16*x*16^k/(8*x*4^k +1/Q(k+1)))))); (continued fraction). - _Sergei N. Gladkovskii_, May 21 2013
%F a(n) = 2*A003674(n-1). - _G. C. Greubel_, Feb 17 2023
%t Table[2^(2n-2)-(-2)^(n-1),{n,30}] (* or *) LinearRecurrence[{2,8},{0,6},30] (* _Harvey P. Dale_, Dec 10 2012 *)
%o (Magma) [4^(n-1)-(-2)^(n-1): n in [1..40]]; // _G. C. Greubel_, Feb 17 2023
%o (SageMath) [4^(n-1)-(-2)^(n-1) for n in range(1,41)] # _G. C. Greubel_, Feb 17 2023
%Y Cf. A003674, A003683.
%K nonn,easy
%O 1,2
%A _John W. Layman_ and Jamaine Paddyfoot (jay_paddyfoot(AT)hotmail.com), Jun 14 2002
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