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 A165961 Number of circular permutations of length n without 3-sequences. 14
 1, 5, 20, 102, 627, 4461, 36155, 328849, 3317272, 36757822, 443846693, 5800991345, 81593004021, 1228906816941, 19733699436636, 336554404751966, 6075478765948135, 115734570482611885, 2320148441078578447, 48827637296350480457, 1076313671861962141616 (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 Michael De Vlieger, Table of n, a(n) for n = 3..450 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, Thesis, 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_{k=0..n/2} binomial(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 MATHEMATICA a[n_] := Sum[Binomial[n-k, k] Subfactorial[n-k-1], {k, 0, n/2}]; a /@ Range[3, 21] (* Jean-François Alcover, Oct 29 2019 *) CROSSREFS Cf. A002628, A165960, A165962. Cf. A000166, A180186, - Emeric Deutsch, Sep 07 2010 A column of A216718. - N. J. A. Sloane, Sep 15 2012 Sequence in context: A110595 A369488 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

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Last modified February 22 09:09 EST 2024. Contains 370250 sequences. (Running on oeis4.)