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A174075 Number of circular permutations of length n without modular consecutive triples i,i+2,i+4.

%I A174075 #8 Feb 06 2024 09:35:00
%S A174075 1,6,18,93,600,4320,35168,321630,3257109,36199458,438126986,
%T A174075 5736774869,80808984725,1218563192160,19587031966352,334329804180135,
%U A174075 6039535339644630,115118210695441900,2308967760171049528,48613722701440862328,1072008447320752890459
%N A174075 Number of circular permutations of length n without modular consecutive triples i,i+2,i+4.
%C A174075 Circular permutations are permutations whose indices are from the ring of integers modulo n.
%H A174075 Max Alekseyev, <a href="/A174075/b174075.txt">Table of n, a(n) for n = 3..100</a>
%H A174075 Wayne M. Dymacek, Isaac Lambert and Kyle Parsons, <a href="http://math.ku.edu/~ilambert/CN.pdf">Arithmetic Progressions in Permutations</a>, 2012.
%F A174075 a(n) = A165962(n) for odd n.
%e A174075 Since a(5)=18, there are (5-1)!-18=4 circular permutations with modular consecutive triples i,i+2,i+4 in all circular permutations of length 5. These are exactly (0,2,4,1,3), (0,2,4,3,1), (0,4,2,1,3), and (0,3,2,4,1). Note some have more than one modular progression.
%t A174075 f[i_,n_,k_]:=If[i==0 && k==0,1,If[i==n && n==k,1,Binomial[k-1,k-i]*Binomial[n-k-1,k-i-1] + 2*Binomial[k-1,k-i-1]*Binomial[n-k-1,k-i-1]+Binomial[k-1,k-i-1]*Binomial[n-k-1,k-i]]];
%t A174075 w1[i_,n_,k_]:=If[n-2k+i<0,0,If[n-2k+i==0,1,(n-2k+i-1)!]];
%t A174075 a[n_,k_]:=Sum[f[i,n,k]*w1[i,n,k],{i,0,k}];
%t A174075 A165962[n_]:=(n-1)!+Sum[(-1)^k*a[n,k],{k,1,n}];
%t A174075 b[n_,k_]:=Sum[Sum[Sum[f[j,n/2,p]*f[i-j,n/2,k-p]*w2[i,j,n,k,p],{p,0,k}],{j,0,i}],{i,0,k-1}];
%t A174075 w2[i_,j_,n_,k_,p_]:=If[n/2-2p+j<=0 || n/2-2(k-p)+(i-j)<=0,0,(n-2k+i-1)!];
%t A174075 A216727[n_?EvenQ]:=(n-1)!+Sum[(-1)^k*b[n,k],{k,1,n}];
%t A174075 A216727[n_?OddQ]:=A165962[n];
%t A174075 Table[A216727[n],{n,3,23}] (* _David Scambler_, Sep 18 2012 *)
%Y A174075 Column 1 of A216726.
%Y A174075 Cf. A165962, A174072, A174073, A174074.
%K A174075 nonn
%O A174075 3,2
%A A174075 _Isaac Lambert_, Mar 06 2010