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A358653 a(n) is the number of trivial braids on 3 strands which are products of n generators a, b, where a = sigma_1 sigma_2 sigma_1 and b = sigma_1 sigma_2.

%I A358653 #12 Jan 16 2024 16:57:41
%S A358653 1,0,4,0,28,10,244,210,2412,3366,26014,49456,299452,701818,3624478
%N A358653 a(n) is the number of trivial braids on 3 strands which are products of n generators a, b, where a = sigma_1 sigma_2 sigma_1 and b = sigma_1 sigma_2.
%C A358653 In the discussion of A354602, _Andrey Zabolotskiy_ asked what the values of the sequence would be if expressed in terms of a and b. This sequence lists these values.
%H A358653 Wikipedia, <a href="https://en.wikipedia.org/wiki/Braid_group#Relation_with_symmetric_group_and_the_pure_braid_group">Relation with symmetric group and the pure braid group</a>.
%o A358653 (SageMath)
%o A358653 B.<s1,s2> = BraidGroup(3)
%o A358653 gen = [s1*s2*s1, s1*s2]
%o A358653 gen += [x^-1 for x in gen]
%o A358653 e = B(())
%o A358653 words, a = {e: 1}, [1]
%o A358653 for n in range(15):
%o A358653     old_words, words = words, {}
%o A358653     for w, c in old_words.items():
%o A358653         for g in gen:
%o A358653             nw = w*g
%o A358653             words[nw] = words.get(nw, 0) + c
%o A358653     a.append(words.get(e, 0))
%o A358653 print(a) # _Andrey Zabolotskiy_, Jan 16 2024
%Y A358653 Cf. A354602.
%K A358653 nonn,more
%O A358653 0,3
%A A358653 _Alexei Vernitski_, Nov 25 2022
%E A358653 a(11)-a(14) from _Andrey Zabolotskiy_, Jan 16 2024