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      Growth behaviour of periodic tame friezes

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          Abstract

          We examine the growth behaviour of the entries occurring in (the lattice of) an n-periodic tame frieze F of real numbers. We begin by observing that the difference between vertically adjacent entries in (non-trivial) rows n-2 and n is constant. Entries separated by intervals of n positions in any given diagonal are then shown to be recursively related, with this constant value appearing as a coefficient in the linear recursion relation. For this reason, we term it a growth coefficient for F. Since n can be any multiple of the minimal period of F, this is one of a family of such coefficients, which are in-turn also recursively related. Analysis of this latter recursion plays an important role in our investigations. We place special emphasis on periodic tame friezes of positive integers, specifying the values the growth coefficients take for any such frieze. We establish that the growth coefficients of the pair of friezes arising from a triangulation of an annulus can be considered to coincide. The entries of both are shown to grow asymptotically exponentially, while triangulations of a punctured disc are seen to provide the only friezes of linear growth.

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          Most cited references11

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          Cluster algebras as Hall algebras of quiver representations

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            Triangulated polygons and frieze patterns

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              Frieze patterns

              H. Coxeter (1971)
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                Author and article information

                Journal
                2016-03-07
                2016-03-17
                Article
                1603.02127
                98c7ccac-e045-46d4-b637-d4a4a3d1b91a

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                05B99, 39A70, 82B20
                Second version with two new remarks and some additional minor changes; 33 pages, 10 figures
                math.CO

                Combinatorics
                Combinatorics

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