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

3,2

Comments

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.

References

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

Links

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

Formula

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

Example

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).

Mathematica

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 *)

Crossrefs

Cf. A002628, A165960, A165961.

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

Sequence in context: A113023 A174082 A188329 * A127756 A228614 A158455

Adjacent sequences: A165959 A165960 A165961 * A165963 A165964 A165965

Keyword

nonn

Author

Isaac Lambert, Oct 01 2009

Status

approved