

A165961


Number of circular permutations of length n without 3sequences.


13



1, 5, 20, 102, 627, 4461, 36155, 328849, 3317272, 36757822, 443846693, 5800991345, 81593004021, 1228906816941, 19733699436636, 336554404751966, 6075478765948135, 115734570482611885, 2320148441078578447
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OFFSET

3,2


COMMENTS

Circular permutations are permutations whose indices are from the ring of integers modulo n. 3sequences are of the form i,i+1,i+2. Sequence gives number of permutations of [n] starting with 1 and having no 3sequences.
a(n) is also the number of permutations of length n1 without consecutive fixed points (cf. A180187).  David Scambler, Mar 27 2011


REFERENCES

Wayne M. Dymacek, Isaac Lambert and Kyle Parsons, Arithmetic Progressions in Permutations, http://math.ku.edu/~ilambert/CN.pdf, 2012.  From N. J. A. Sloane, Sep 15 2012 [broken link]


LINKS

Table of n, a(n) for n=3..21.
Wayne M. Dymacek and Isaac Lambert, Permutations Avoiding Runs of i, i+1, i+2 or i, i1, i2, Journal of Integer Sequences, Vol. 14 (2011), Article 11.1.6
Kyle Parsons, Arithmetic progressions in permutations, 2011


FORMULA

Let b(n) be the sequence A002628. Then for n>5, this sequence satisfies a(n)=b(n1)b(n3)+a(n3).
a(n) = sum(binom(nk,k)*d(nk1), where d(j)=A000166(j) are the derangement numbers.  Emeric Deutsch, Sep 07 2010


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).


MAPLE

d[0] := 1: for n to 51 do d[n] := n*d[n1]+(1)^n end do: a := proc (n) options operator, arrow: sum(binomial(nk, k)*d[nk1], k = 0 .. floor((1/2)*n)) end proc: seq(a(n), n = 3 .. 23); # Emeric Deutsch, Sep 07 2010


CROSSREFS

Cf. A002628, A165960, A165962.
Cf. A000166, A180186 [From Emeric Deutsch, Sep 07 2010]
A column of A216718.  From N. J. A. Sloane, Sep 15 2012
Sequence in context: A108509 A110595 A092640 * A276314 A292358 A259275
Adjacent sequences: A165958 A165959 A165960 * A165962 A165963 A165964


KEYWORD

nonn


AUTHOR

Isaac Lambert, Oct 01 2009


EXTENSIONS

More terms from Emeric Deutsch, Sep 07 2010
Edited by N. J. A. Sloane, Apr 04 2011


STATUS

approved



