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A033305 Number of permutations (p1,...,pn) such that 1<=|pk-k|<=2 for all k. 7
1, 0, 1, 2, 4, 6, 13, 24, 45, 84, 160, 300, 565, 1064, 2005, 3774, 7108, 13386, 25209, 47472, 89401, 168360, 317056, 597080, 1124425, 2117520, 3987721, 7509690, 14142276, 26632782, 50154949, 94451976, 177872293 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

REFERENCES

Lehmer, D. H.; Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.

R. P. Stanley, Enumerative Combinatorics I, p. 252, Example 4.7.16.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties , arXiv:1103.2582 [math.CO], 2011-2013.

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,-1).

FORMULA

G.f.: (1-x)/( (1+x)*(x^4-2*x^3+x^2-2*x+1) ).

a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) - a(n-5).

a(n) = h(n)-h(n-1), n>0, h(n) = sum(sum(C(k,r)*sum(C(r,m)*sum(C(m,j)*C(j,n-m-k-j-r) *(-1)^(n-m-k-j-r), j=0..m), m=0..r), r=0..k), k=1..n). - Vladimir Kruchinin, Sep 10 2010

Lim_{n ->Inf} a(n)/a(n-1) = (1 + sqrt(2) + sqrt(2*sqrt(2)-1)) /2 = 1.88320350591... for n>2. Lim_{n ->Inf} a(n-1)/a(n) = (1 + sqrt(2) - sqrt(2*sqrt(2)-1)) /2 = .53101005645... for n>0. [Tim Monahan, Aug 09 2011]

7*a(n) = 2*(-1)^n -8*A112575(n) -2*A112575(n-2) +6*A112575(n-1) +5*A112575(n+1). - R. J. Mathar, Sep 27 2013

Empirical: a(n)+a(n+1) = A183324(n). - R. J. Mathar, Sep 27 2013

MATHEMATICA

LinearRecurrence[{1, 1, 1, 1, -1}, {1, 0, 1, 2, 4}, 40] (* Harvey P. Dale, Aug 28 2012 *)

PROG

(Maxima) h(n) := sum(sum(binomial(k, r) *sum(binomial(r, m) *sum(binomial(m, j) *binomial(j, n-m-k-j-r) *(-1)^(n-m-k-j-r), j, 0, m), m, 0, r), r, 0, k), k, 1, n); a(n):=h(n)-h(n-1); \\ Vladimir Kruchinin, Sep 10 2010

CROSSREFS

Column k=2 of A259776.

Cf. A260074.

Sequence in context: A109078 A291738 A321228 * A105543 A295618 A027712

Adjacent sequences:  A033302 A033303 A033304 * A033306 A033307 A033308

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

New description from Max Alekseyev, Jul 09 2006

STATUS

approved

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Last modified November 19 10:48 EST 2018. Contains 317349 sequences. (Running on oeis4.)