A216723 -id:A216723 - cp's OEIS frontend

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

1, 5, 18, 95, 600, 4307, 35168, 321609, 3257109, 36199762, 438126986, 5736774126, 80808984725, 1218563180295, 19587031966352, 334329804347219, 6039535339644630, 115118210694558105, 2308967760171049528, 48613722701436777455, 1072008447320752890459

Offset: 3

Isaac Lambert, Oct 01 2009

nonn

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.

			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).

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

Max Alekseyev, Table of n, a(n) for n = 3..100

Cf. A002628, A165960, A165961.

First column of A216722. Cf. A216723. - N. J. A. Sloane, Sep 15 2012

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]]];
w1[i_,n_,k_]:=If[n-2k+i<0,0,If[n-2k+i==0,1,(n-2k+i-1)!]];
a[n_,k_]:=Sum[f[i,n,k]*w1[i,n,k],{i,0,k}];
A165962[n_]:=(n-1)!+Sum[(-1)^k*a[n,k],{k,1,n}];
Table[A165962[n],{n,3,23}] (* David Scambler, Sep 18 2012 *)

This sequence can be related to A165961 by the use of auxiliary sequences (and the auxiliary sequences can themselves be calculated by recurrence relations).

A216722 Triangle read by rows: number of circular permutations of [1..n] with k modular progressions of rise 1, distance 1 and length 3 (n >= 3, 0 <= k <= n).

Original entry on oeis.org

1, 0, 0, 1, 5, 0, 0, 0, 1, 18, 5, 0, 0, 0, 1, 95, 18, 6, 0, 0, 0, 1, 600, 84, 28, 7, 0, 0, 0, 1, 4307, 568, 116, 40, 8, 0, 0, 0, 1, 35168, 4122, 810, 156, 54, 9, 0, 0, 0, 1, 321609, 33910, 5975, 1100, 205, 70, 10, 0, 0, 0, 1

Offset: 3

N. J. A. Sloane, Sep 15 2012

			Triangle begins:
       1,     0,    0,    1;
       5,     0,    0,    0,   1;
      18,     5,    0,    0,   0,  1;
      95,    18,    6,    0,   0,  0,  1;
     600,    84,   28,    7,   0,  0,  0, 1;
    4307,   568,  116,   40,   8,  0,  0, 0, 1;
   35168,  4122,  810,  156,  54,  9,  0, 0, 0, 1;
  321609, 33910, 5975, 1100, 205, 70, 10, 0, 0, 0, 1;
  ...

Wayne M. Dymacek, Isaac Lambert and Kyle Parsons, Arithmetic Progressions in Permutations, http://math.ku.edu/~ilambert/CN.pdf, 2012.

Columns 1..2 are A165962, A216723.
Row sums are A000142(n-1).
Cf. A216716, A216718, A216719, A165962.

cp's OEIS Frontend

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

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