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 A165962 Number of circular permutations of length n without modular 3-sequences 15
 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 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 August 7 10:19 EDT 2022. Contains 355985 sequences. (Running on oeis4.)