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A165962 Number of circular permutations of length n without modular 3-sequences 16
1, 5, 18, 95, 600, 4307, 35168, 321609, 3257109, 36199762, 438126986, 5736774126, 80808984725, 1218563180295, 19587031966352, 334329804347219, 6039535339644630, 115118210694558105, 2308967760171049528, 48613722701436777455, 1072008447320752890459 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=3..23.

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

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Last modified July 5 03:37 EDT 2020. Contains 335459 sequences. (Running on oeis4.)