A071930 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.

0, 6, 12, 72, 240, 1056, 4032, 16512, 65280, 262656, 1047552, 4196352, 16773120, 67117056, 268419072, 1073774592, 4294901760, 17180000256, 68719214592, 274878431232, 1099510579200, 4398048608256, 17592181850112

Offset

1,2

Comments

a(n) = A003683(n+1)/6. No words of odd length (see the description above) reduce to 1.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,8).

Formula

a(n) = 2^(2*n-2) - (-2)^(n-1) = 6*A003683(n-1).

From Harvey P. Dale, Dec 10 2012: (Start)

a(n) = 2*a(n-1) + 8*a-(n-2).

G.f.: 6*x/(1-2*x-8*x^2)). (End)

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

a(n) = 2*A003674(n-1). - G. C. Greubel, Feb 17 2023

Mathematica

Table[2^(2n-2)-(-2)^(n-1), {n, 30}] (* or *) LinearRecurrence[{2, 8}, {0, 6}, 30] (* Harvey P. Dale, Dec 10 2012 *)

Prog

(Magma) [4^(n-1)-(-2)^(n-1): n in [1..40]]; // G. C. Greubel, Feb 17 2023
(SageMath) [4^(n-1)-(-2)^(n-1) for n in range(1, 41)] # G. C. Greubel, Feb 17 2023

Crossrefs

Cf. A003674, A003683.

Sequence in context: A088726 A163342 A107904 * A372180 A239854 A061520

Adjacent sequences: A071927 A071928 A071929 * A071931 A071932 A071933

Keyword

nonn,easy

Author

John W. Layman and Jamaine Paddyfoot (jay_paddyfoot(AT)hotmail.com), Jun 14 2002

Status

approved