cp's OEIS Frontend

A111340 Number of positive integer 2-friezes with n-1 nontrivial rows.

%I A111340 #25 May 22 2025 04:39:22
%S A111340 1,5,51,868,26952
%N A111340 Number of positive integer 2-friezes with n-1 nontrivial rows.
%C A111340 The n-th term is the number of positive integer tables a(i,m) (with i running from 1 to n+3 and m running from minus infinity to infinity) subject to the boundary conditions a(i,m) = 0 when i = 1 or i = n+3 and a(i,m) = 1 when i = 2 or i = n+2 and the internal condition a(i,m-1) a(i,m+1) = a(i-1,m) a(i+1,m) + a(i,m) when i is strictly between 2 and n+2.
%C A111340 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 A111340 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. - _Andrei Zabolotskii_, Nov 01 2022
%H A111340 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.
%H A111340 Robin Zhang, <a href="https://arxiv.org/abs/2503.08800">A positive Siegel theorem: Dynkin friezes and positive Mordell-Schinzel</a>, arXiv:2503.08800 [math.NT], 2025.
%e A111340 The number 1 in the sequence is counting the rather boring configuration
%e A111340     0 0 0 0 0 0 0 0
%e A111340 ... 1 1 1 1 1 1 1 1 ...
%e A111340     1 1 1 1 1 1 1 1
%e A111340     0 0 0 0 0 0 0 0
%e A111340 The number 5 is counting the configuration
%e A111340     0 0 0 0 0 0 0 0 0 0
%e A111340     1 1 1 1 1 1 1 1 1 1
%e A111340 ... 1 1 2 3 2 1 1 2 3 2 ...
%e A111340     1 1 1 1 1 1 1 1 1 1
%e A111340     0 0 0 0 0 0 0 0 0 0
%e A111340 and its four distinct cyclic shifts, each of which repeats with period 5 (note the Lyness 5-cycle A076839 in the middle).
%e A111340 a(2) = A000108(3) = number of friezes of type A_2 (cyclic shifts of A139434), a(3) = A247415(4). a(4) and a(5) also count friezes of types resp. E_6 and E_8.
%Y A111340 Cf. A000108, A247415, A076839, A139434.
%K A111340 nonn,fini,full
%O A111340 1,2
%A A111340 _N. J. A. Sloane_, based on correspondence from _James Propp_, May 08 2005
%E A111340 The last finite term, a(5), added based on Zhang's preprint and name clarified by _Andrei Zabolotskii_, May 14 2025