This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A199044 #30 Mar 24 2017 00:47:54
%S A199044 1,0,6,0,74,0,1140,0,19562,0,357756,0,6824684,0,134166696,0,
%T A199044 2697855082,0,55213424556,0,1146078241284,0,24067465856088,0,
%U A199044 510351502965548,0,10911807871502232,0,234970037988773560,0,5091149074269149520,0,110912377099411850090,0
%N A199044 The number of identity elements of length n in Z*Z^2.
%C A199044 Z*Z^2 is the free product of the free group on one letter (say, x) and the free abelian group on two letters (say, y and z).
%C A199044 Viewed as the quotient of the free group F on three letters {x,y,z} by the normal subgroup generated by the commutator [y,z], the sequence gives the number of words in F of length n that are sent to the identity in Z*Z^2 under the quotient map.
%C A199044 Note that odd-numbered terms are zero.
%D A199044 Derek F. Holt, Sarah Rees, Claas E. Röver, and Richard M. Thomas, Groups with Context-Free Co-Word Problem, J. London Math. Soc. (2005) 71 (3): 643-657. doi: 10.1112/S002461070500654X
%D A199044 Brough, Tara Rose, Groups with poly-context-free word problem, PhD thesis (2010), University of Warwick.
%H A199044 Nick Loughlin, <a href="/A199044/b199044.txt">Table of n, a(n) for n = 0..881</a>
%e A199044 The identity from the free group F maps to the identity in Z*Z^2, and is the only word of length zero in F, so a(0)=1.
%e A199044 The group Z*Z^2 maps onto the direct product C_2^3, the group of exponent 2 with 8 elements. Therefore no elements of odd length are sent to the identity and thus a(2i-1)=0 for all positive integers i.
%e A199044 The only word of length zero is the empty word, which vacuously represents the identity. Therefore, a_0=1.
%e A199044 For n=2, there are a_2=6 identities; each is a (positive or negative) generator x,y, or z, followed or preceded by its inverse. We have the words x*x^-1, y*y^-1, z*z^-1, plus the reverse of each.
%K A199044 nonn,word
%O A199044 0,3
%A A199044 _Nick Loughlin_, Nov 02 2011
%E A199044 Edited by _Max Alekseyev_, Jan 24 2012
%E A199044 Edited by Nick Loughlin, Mar 12 2012