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A199044
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The number of identity elements of length n in Z*Z^2.
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1
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1, 0, 6, 0, 74, 0, 1140, 0, 19562, 0, 357756, 0, 6824684, 0, 134166696, 0, 2697855082, 0, 55213424556, 0, 1146078241284, 0, 24067465856088, 0, 510351502965548, 0, 10911807871502232, 0, 234970037988773560, 0, 5091149074269149520, 0, 110912377099411850090, 0
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OFFSET
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0,3
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COMMENTS
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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.
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REFERENCES
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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
Brough, Tara Rose, Groups with poly-context-free word problem, PhD thesis (2010), University of Warwick.
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LINKS
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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nonn,word
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AUTHOR
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EXTENSIONS
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Edited by Nick Loughlin, Mar 12 2012
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STATUS
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approved
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