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%I #10 Feb 06 2024 09:27:05
%S 1,5,18,95,600,4307,35168,321609,3257109,36199762,438126986,
%T 5736774126,80808984725,1218563180295,19587031966352,334329804347219,
%U 6039535339644630,115118210694558105,2308967760171049528,48613722701436777455,1072008447320752890459
%N Number of circular permutations of length n without modular 3-sequences
%C Circular permutations are permutations whose indices are from the ring of integers modulo n. Modular 3-sequences are of the following form: i,i+1,i+2, where arithmetic is modulo n.
%D Wayne M. Dymacek, Isaac Lambert and Kyle Parsons, Arithmetic Progressions in Permutations, http://math.ku.edu/~ilambert/CN.pdf, 2012. - _N. J. A. Sloane_, Sep 15 2012
%H Max Alekseyev, <a href="/A165962/b165962.txt">Table of n, a(n) for n = 3..100</a>
%F This sequence can be related to A165961 by the use of auxiliary sequences (and the auxiliary sequences can themselves be calculated by recurrence relations).
%e For n=4 the a(4)=5 solutions are (0,1,3,2), (0,2,1,3), (0,2,3,1), (0,3,1,2) and (0,3,2,1).
%t 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 w1[i_,n_,k_]:=If[n-2k+i<0,0,If[n-2k+i==0,1,(n-2k+i-1)!]];
%t a[n_,k_]:=Sum[f[i,n,k]*w1[i,n,k],{i,0,k}];
%t A165962[n_]:=(n-1)!+Sum[(-1)^k*a[n,k],{k,1,n}];
%t Table[A165962[n],{n,3,23}] (* _David Scambler_, Sep 18 2012 *)
%Y Cf. A002628, A165960, A165961.
%Y First column of A216722. Cf. A216723. - _N. J. A. Sloane_, Sep 15 2012
%K nonn
%O 3,2
%A _Isaac Lambert_, Oct 01 2009
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