cp's OEIS Frontend

A165962 Number of circular permutations of length n without modular 3-sequences.

%I A165962 #12 Jul 22 2025 07:23:58
%S A165962 1,5,18,95,600,4307,35168,321609,3257109,36199762,438126986,
%T A165962 5736774126,80808984725,1218563180295,19587031966352,334329804347219,
%U A165962 6039535339644630,115118210694558105,2308967760171049528,48613722701436777455,1072008447320752890459
%N A165962 Number of circular permutations of length n without modular 3-sequences.
%C A165962 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 A165962 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 A165962 Max Alekseyev, <a href="/A165962/b165962.txt">Table of n, a(n) for n = 3..100</a>
%F A165962 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 A165962 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 A165962 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 A165962 w1[i_,n_,k_]:=If[n-2k+i<0,0,If[n-2k+i==0,1,(n-2k+i-1)!]];
%t A165962 a[n_,k_]:=Sum[f[i,n,k]*w1[i,n,k],{i,0,k}];
%t A165962 A165962[n_]:=(n-1)!+Sum[(-1)^k*a[n,k],{k,1,n}];
%t A165962 Table[A165962[n],{n,3,23}] (* _David Scambler_, Sep 18 2012 *)
%Y A165962 Cf. A002628, A165960, A165961.
%Y A165962 First column of A216722. Cf. A216723. - _N. J. A. Sloane_, Sep 15 2012
%K A165962 nonn
%O A165962 3,2
%A A165962 _Isaac Lambert_, Oct 01 2009