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A232864
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Number of permutations of n elements not cyclically containing the consecutive pattern 123.
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2
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1, 1, 2, 3, 12, 45, 234, 1323, 8856, 65529, 543510, 4937031, 49030596, 526930677, 6101871426, 75686176035, 1001517264432, 14079895613937, 209594037600558, 3293305758743679, 54470994630103260, 945988795762018029, 17211193919411902938, 327371367293394753627
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OFFSET
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0,3
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..200
R. Ehrenborg, Cyclically consecutive permutation avoidance, arXiv:1312.2051 [math.CO], 2013
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FORMULA
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a(n) = n! * Sum_{k=-oo..oo} (sqrt(3)/(2*Pi*(k+1/3)))^n for n >= 2.
a(n) = A080635(n-1)*n for n>0. - Alois P. Heinz, Dec 01 2013
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EXAMPLE
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For n=4 the a(4) = 12 comes from the 3 permutations 1324, 1423 and 1432; and by cyclically shifting we obtain 3 * 4 = 12 permutations.
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MAPLE
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b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
`if`(t<2, add(b(u+j-1, o-j, t+1), j=1..o), 0)+
add(b(u-j, o+j-1, 1), j=1..u))
end:
a:= n-> `if`(n=0, 1, n*b(0, n-1, 1)):
seq(a(n), n=0..25); # Alois P. Heinz, Dec 01 2013
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MATHEMATICA
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b[u_, o_, t_] := b[u, o, t] = If[u+o==0, 1, If[t<2, Sum[b[u+j-1, o-j, t+1], {j, 1, o}], 0] + Sum[b[u-j, o+j-1, 1], {j, 1, u}]];
a[n_]:= If[n==0, 1, n*b[0, n-1, 1]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 14 2017, after Alois P. Heinz *)
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CROSSREFS
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Cf. A049774, A080635.
Sequence in context: A012306 A012312 A009243 * A307957 A307956 A358716
Adjacent sequences: A232861 A232862 A232863 * A232865 A232866 A232867
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KEYWORD
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nonn
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AUTHOR
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Richard Ehrenborg, Dec 01 2013
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STATUS
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approved
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