A396638 - OEIS

%I #9 Jun 04 2026 10:16:59

%S 1,4,12,36,96,248,624,1548,3804,9292,22608,54864,132912,321620,777660,

%T 1879380,4540368,10966504,26483712,63950652,154412076,372818588,

%U 900120096,2173173408,5246652384,12666778276,30580694508,73828952964,178239871680,430310753240,1038864706320,2508045550956

%N Number of geodesics of length n in the braid group B_3.

%H Lucas Sabalka, <a href="https://arxiv.org/abs/math/0311153">Geodesics in the braid group on three strands</a>; arXiv:math/0311153 [math.GR], 2003.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-3,-1).

%F G.f.: (x^4+3*x^3+x+1)/((x^2+x-1)*(x^2+2*x-1)).

%e Let a,b be the generators of B_3. The a(3) = 36 geodesics of length 3 in B_3 are:

%e     a^2*b^-1,           b^-2*a,         b^2*a^-1,         a^-1*b^2,            b^2*a,

%e          b^3,         b^-1*a^2,           b*a^-2,         b^-2*a^-1,     b^-1*a*b^-1,

%e        a*b^2,        b^-1*a^-2,             b^-3,            a^2*b,           a*b^-2,

%e  a^-1*b*a^-1,        a^-1*b^-2,         b*a^-1*b,             a^-3,        a^-2*b^-1,

%e        b*a^2,              a^3,         a*b^-1*a,           a^-2*b,

%e   a*b*a    =  b*a*b,           a^-1*b^-1*a^-1 =  b^-1*a^-1*b^-1,

%e   a*b*a^-1 =  b^-1*a*b,        b^-1*a^-1*b    =  a*b^-1*a^-1,

%e   b*a*b^-1 =  a^-1*b*a,        a^-1*b^-1*a    =  b*a^-1*b^-1.

%t LinearRecurrence[{3,0,-3,-1},{1,4,12,36,96},32] (* _Stefano Spezia_, Jun 01 2026 *)

%Y Cf. A000079 (geodesics in the braid monoid), A396637 (elements of geodesic length).

%K nonn,easy

%O 0,2

%A _Ludovic Schwob_, Jun 01 2026