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A111340
Related to frieze patterns.
0
1, 5, 51, 868
OFFSET
1,2
COMMENTS
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).
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
EXAMPLE
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 A076839 down the middle).
CROSSREFS
Sequence in context: A077392 A193444 A243242 * A124559 A368838 A348023
KEYWORD
nonn,more,fini
AUTHOR
N. J. A. Sloane, based on correspondence from James Propp, May 08 2005
STATUS
approved