A396637 Number of elements of geodesic length n in the braid group B_3.

1, 4, 12, 30, 68, 148, 314, 656, 1356, 2782, 5676, 11532, 23354, 47176, 95108, 191438, 384852, 772900, 1550970, 3110304, 6234140, 12490174, 25015772, 50088860, 100270458, 200690968, 401624724, 803642286, 1607920196, 3216868852, 6435401786, 12873496112, 25751348844, 51509746846, 103030899468

Offset

0,2

Links

Table of n, a(n) for n=0..34.

Lucas Sabalka, Geodesics in the braid group on three strands; arXiv:math/0311153 [math.GR], 2003.

Index entries for linear recurrences with constant coefficients, signature (4,-4,-1,2).

Formula

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

Example

Let a,b be the generators of B_3. The a(3) = 30 elements of geodesic length 3 in B_3 are:
    a^2*b^-1,           b^-2*a,         b^2*a^-1,         a^-1*b^2,            b^2*a,
         b^3,         a*b*a^-1,         b^-1*a^2,         b*a*b^-1,           b*a^-2,
   b^-2*a^-1,   a^-1*b^-1*a^-1,      b^-1*a*b^-1,            a*b^2,        b^-1*a^-2,
       a*b*a,             b^-3,            a^2*b,      a^-1*b^-1*a,           a*b^-2,
 a^-1*b*a^-1,        a^-1*b^-2,         b*a^-1*b,             a^-3,        a^-2*b^-1,
 b^-1*a^-1*b,            b*a^2,              a^3,         a*b^-1*a,           a^-2*b.

Mathematica

LinearRecurrence[{4, -4, -1, 2}, {1, 4, 12, 30, 68}, 35] (* Stefano Spezia, Jun 01 2026 *)

Crossrefs

Cf. A000071 (elements of geodesic length in the braid monoid), A396638 (geodesics).

Sequence in context: A338223 A118425 A097809 * A272144 A036389 A037166

Adjacent sequences: A396634 A396635 A396636 * A396638 A396639 A396640

Keyword

nonn,easy

Author

Ludovic Schwob, Jun 01 2026

Status

approved