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%I #17 Dec 12 2022 18:17:27
%S 1,5,51,868
%N Related to frieze patterns.
%C The n-th term is the number of positive integer tables a(i,n) (with i running from 1 to n+3 and n running from minus infinity to infinity) subject to the boundary conditions a(i,n) = 0 when i = 1 or i = n+3 and a(i,n) = 1 when i = 2 or i = n+2 and the internal condition a(i,n-1) a(i,n+1) = a(i-1,n) a(i+1,n) + a(i,n) when i is strictly between 2 and n+2.
%C It is not known as of this writing whether any or all of the terms of the sequence beyond 868 are finite. If the final term "a(i,n)" in the internal condition is replaced by "1", then what we are looking is just a frieze pattern a la Conway and Coxeter (or rather two interlaced frieze patterns that do not interact at all).
%C According to the lecture notes by S. Morier-Genoud (see paragraph "2-frieze of positive integers"), a(5) is conjectured to be 26952, and it is proved that there are no more finite terms. - _Andrey Zabolotskiy_, Nov 01 2022
%H Sophie Morier-Genoud, <a href="https://sophie-moriergenoud.perso.math.cnrs.fr/Leicester_Lectures_Exercises.pdf">Lecture notes on Integrable Systems and Friezes</a>, 2017.
%e The number 1 in the sequence is counting the rather boring configuration
%e ...
%e 0 1 1 0
%e 0 1 1 0
%e 0 1 1 0
%e 0 1 1 0
%e 0 1 1 0
%e 0 1 1 0
%e 0 1 1 0
%e 0 1 1 0
%e ...
%e The number 5 is counting the configuration
%e ...
%e 0 1 1 1 0
%e 0 1 1 1 0
%e 0 1 2 1 0
%e 0 1 3 1 0
%e 0 1 2 1 0
%e 0 1 1 1 0
%e 0 1 1 1 0
%e 0 1 2 1 0
%e 0 1 3 1 0
%e 0 1 2 1 0
%e ...
%e and its four distinct cyclic shifts, each of which repeats with period 5 (note the Lyness 5-cycle A076839 down the middle).
%K nonn,more,fini
%O 1,2
%A _N. J. A. Sloane_, based on correspondence from _James Propp_, May 08 2005