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COMMENT
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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.
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).
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EXAMPLE
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The number 1 in the sequence is counting the rather boring configuration
...
0 1 1 0
0 1 1 0
0 1 1 0
0 1 1 0
0 1 1 0
0 1 1 0
0 1 1 0
0 1 1 0
...
The number 5 is counting the configuration
...
0 1 1 1 0
0 1 1 1 0
0 1 2 1 0
0 1 3 1 0
0 1 2 1 0
0 1 1 1 0
0 1 1 1 0
0 1 2 1 0
0 1 3 1 0
0 1 2 1 0
...
and its four distinct cyclic shifts, each of which repeats with period 5
(note the Lyness 5-cycle down the middle).
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