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Number of total cyclic orders Z on {0, ..., n-1} such that (i, (i+1) mod n, (i+2) mod n) in Z for 0 <= i < n.
4

%I #39 Oct 31 2022 19:17:23

%S 1,1,1,1,2,9,31,128,708,4015,24865,177444,1357830,11141634,99680595,

%T 953369248,9687797896,104909705019,1202985610821,14542305462860,

%U 185165060805578,2476043157780478,34673872424313463

%N Number of total cyclic orders Z on {0, ..., n-1} such that (i, (i+1) mod n, (i+2) mod n) in Z for 0 <= i < n.

%C For all n >= 1, a(n) is the number of n-sided polygons, turning always in the same direction (right or left) while following its edges. By "n-sided polygons" we mean the polygons that can be drawn by connecting n equally spaced points on a circle. - _Ludovic Schwob_, Apr 04 2021

%C For all n >= 1, a(n) is the number of cyclic permutations of length n that avoid consecutive patterns 123, 231, and 312. - _Rupert Li_, Jul 27 2021

%H Ludovic Schwob, <a href="/A295264/b295264.txt">Table of n, a(n) for n = 1..39</a>

%H Rupert Li, <a href="https://arxiv.org/abs/2107.12353">Vincular Pattern Avoidance on Cyclic Permutations</a>, arXiv:2107.12353 [math.CO], 2021.

%H Sanjay Ramassamy, <a href="https://arxiv.org/abs/1706.03386">Extensions of partial cyclic orders, Euler numbers and multidimensional boustrophedons</a>, arXiv:1706.03386 [math.CO], 2017.

%H Ludovic Schwob, <a href="/A295264/a295264_1.pdf">Illustration of a(7) and a(8)</a>

%o (PARI) \\ Needs B function from A343257.

%o a(n)={sum(i=1, n, B(n,i,1))} \\ _Andrew Howroyd_, May 16 2021

%Y Row sums of A343257.

%K nonn

%O 1,5

%A _Eric M. Schmidt_, Nov 19 2017

%E a(12) corrected and a(13)-a(18) from _Andrew Howroyd_, May 15 2021

%E Corrected initial offset/terms by _Rupert Li_, Sep 17 2021

%E a(19) onwards from _Ludovic Schwob_, Oct 31 2022