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A165961 Number of circular permutations of length n without 3-sequences. 13
1, 5, 20, 102, 627, 4461, 36155, 328849, 3317272, 36757822, 443846693, 5800991345, 81593004021, 1228906816941, 19733699436636, 336554404751966, 6075478765948135, 115734570482611885, 2320148441078578447 (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. 3-sequences are of the form i,i+1,i+2. Sequence gives number of permutations of [n] starting with 1 and having no 3-sequences.

a(n) is also the number of permutations of length n-1 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, i-1, i-2, 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(n-1)-b(n-3)+a(n-3).

a(n) = sum(binom(n-k,k)*d(n-k-1), 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[n-1]+(-1)^n end do: a := proc (n) options operator, arrow: sum(binomial(n-k, k)*d[n-k-1], 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 A259275 A109500

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

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Last modified March 23 19:17 EDT 2017. Contains 283957 sequences.