A396638 Number of geodesics of length n in the braid group B_3.

1, 4, 12, 36, 96, 248, 624, 1548, 3804, 9292, 22608, 54864, 132912, 321620, 777660, 1879380, 4540368, 10966504, 26483712, 63950652, 154412076, 372818588, 900120096, 2173173408, 5246652384, 12666778276, 30580694508, 73828952964, 178239871680, 430310753240, 1038864706320, 2508045550956

Offset

0,2

Links

Table of n, a(n) for n=0..31.

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 (3,0,-3,-1).

Formula

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

Example

Let a,b be the generators of B_3. The a(3) = 36 geodesics of 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,         b^-1*a^2,           b*a^-2,         b^-2*a^-1,     b^-1*a*b^-1,
       a*b^2,        b^-1*a^-2,             b^-3,            a^2*b,           a*b^-2,
 a^-1*b*a^-1,        a^-1*b^-2,         b*a^-1*b,             a^-3,        a^-2*b^-1,
       b*a^2,              a^3,         a*b^-1*a,           a^-2*b,
  a*b*a    =  b*a*b,           a^-1*b^-1*a^-1 =  b^-1*a^-1*b^-1,
  a*b*a^-1 =  b^-1*a*b,        b^-1*a^-1*b    =  a*b^-1*a^-1,
  b*a*b^-1 =  a^-1*b*a,        a^-1*b^-1*a    =  b*a^-1*b^-1.

Mathematica

LinearRecurrence[{3, 0, -3, -1}, {1, 4, 12, 36, 96}, 32] (* Stefano Spezia, Jun 01 2026 *)

Crossrefs

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

Sequence in context: A261584 A347990 A396614 * A002842 A051041 A377170

Adjacent sequences: A396635 A396636 A396637 * A396639 A396640 A396641

Keyword

nonn,easy

Author

Ludovic Schwob, Jun 01 2026

Status

approved