Geodesic
Diophantine representations of linear recurrences. I
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 4, Tome 227 (1995), pp. 52-60 
M. A. Vsemirnov. Diophantine representations of linear recurrences. I. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 4, Tome 227 (1995), pp. 52-60. http://geodesic.mathdoc.fr/item/ZNSL_1995_227_a6/
@article{ZNSL_1995_227_a6,
     author = {M. A. Vsemirnov},
     title = {Diophantine representations of linear {recurrences.~I}},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {52--60},
     year = {1995},
     volume = {227},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1995_227_a6/}
}
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Direct constructions of Diophantine representations of linear recurrent sequences are discussed. These constructions generalize already known results for second-order recurrences. Some connections of this problem with the theory of units in rings of algebraic integers are shown. It is proved that the required representations erist only for second-, third-, and fourth-order sequences. In the two last-mentioned cases certain additional restrictions on their coefficients must be imposed. Bibliography: 14 titles.