OFFSET
0,3
COMMENTS
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).
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.
Note that odd-numbered terms are zero.
REFERENCES
Brough, Tara Rose, Groups with poly-context-free word problem, PhD thesis (2010), University of Warwick.
LINKS
Nick Loughlin, Table of n, a(n) for n = 0..881
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.
EXAMPLE
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.
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.
The only word of length zero is the empty word, which vacuously represents the identity. Therefore, a_0=1.
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.
CROSSREFS
KEYWORD
nonn,word
AUTHOR
Nick Loughlin, Nov 02 2011
EXTENSIONS
Edited by Max Alekseyev, Jan 24 2012
Edited by Nick Loughlin, Mar 12 2012
STATUS
approved