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%I #12 Aug 07 2019 21:12:38
%S 1,1,1,1,2,3,2,2,4,6,2,2,6,8,4,4,6,10,4,4,10,12,4,4,12,18,6,6,8,12,8,
%T 8,16,22,6,6,18,22,8,8,12,22,10,10,22,26,8,8,20,32,12,12,18,24,12,12,
%U 28,36,8,8,30,38,16,16,20,36,16,16,24,30,12,12,36,54
%N Cyclic permutations of length n that avoid the patterns 123 and 231.
%H Miklos Bona, Michael Cory, <a href="https://arxiv.org/abs/1805.05196">Cyclic Permutations Avoiding Pairs of Patterns of Length Three</a>, arXiv:1805.05196 [math.CO], 2018
%F a(2)=1; a(n)=phi(n/2) if n=4k, a(n)=phi((n+2)/4) +phi(n/2) if n=4k+2 > 2, and a(n)=phi((n+1)/2) if n is odd, where phi is the Euler totient function.
%e a(4)=1, since the only cyclic permutation of length 4 avoiding both 123 and 231 is (4231)=4312.
%Y Cf. A000010.
%K nonn
%O 1,5
%A _Miklos Bona_, Aug 07 2019