Geodesic
The number of words of a~given length in the planar crystallographic groups
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 19, Tome 302 (2003), pp. 188-197

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This paper reveals an interrelation between the words in groups and the growth process of the corresponding graphs, based on which exact formulas for the number of words of a given length for all 17 planar crystallographic groups are obtained. Sloane's conjecture on the graphs of these groups is proved. 
@article{ZNSL_2003_302_a12,
     author = {A. V. Shutov},
     title = {The number of words of a~given length in the planar crystallographic groups},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {188--197},
     publisher = {mathdoc},
     volume = {302},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_302_a12/}
}
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A. V. Shutov. The number of words of a~given length in the planar crystallographic groups. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 19, Tome 302 (2003), pp. 188-197. http://geodesic.mathdoc.fr/item/ZNSL_2003_302_a12/