A354602 a(n) is the number of trivial braids on 3 strands with 2*n crossings.

1, 4, 28, 244, 2412, 25804, 290932, 3403404, 40914508, 502307164, 6270949548, 79367274980, 1016035396740, 13133562994244, 171187923980332, 2247551090425204, 29696773531915404, 394599762905224828, 5269737867425481148, 70694195397452450484, 952251900856081556252

Offset

0,2

Comments

In other words, a(n) is the number of products of 2*n generators in the braid group B_3 which are equal to the identity element of the group.

Only braids with an even number of crossings are considered because a braid with an odd number of crossings cannot be trivial.

If we do include the 0s corresponding to the odd values of the number of crossings, a group-theoretical name for this sequence is the cogrowth sequence of B_3.

Links

Table of n, a(n) for n=0..20.

Crossrefs

Cf. A000984 (number of trivial braids on 2 strands with 2*n crossings), A047849 (number of trivial permutations of 3 elements after 2*n adjacent transpositions).

Sequence in context: A199561 A103211 A229644 * A228714 A381913 A371693

Adjacent sequences: A354599 A354600 A354601 * A354603 A354604 A354605

Keyword

nonn

Author

Alexei Vernitski, Jul 08 2022

Extensions

a(9)-a(20) from Christian Sievers, Feb 26 2026

Status

approved