

A084509


Number of groundstate 3ball juggling sequences of period n.


10



1, 1, 2, 6, 24, 96, 384, 1536, 6144, 24576, 98304, 393216, 1572864, 6291456, 25165824, 100663296, 402653184, 1610612736, 6442450944, 25769803776, 103079215104, 412316860416, 1649267441664, 6597069766656, 26388279066624, 105553116266496, 422212465065984
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OFFSET

0,3


COMMENTS

This sequence counts the length n asynchronic site swaps given in A084501/A084502.
Equals row sums of triangle A145463.  Gary W. Adamson, Oct 11 2008
a(n) is the number of permutations of length n+1 avoiding the partially ordered pattern (POP) {1>2, 1>3, 1>4, 1>5} of length 5. That is, the number of length n+1 permutations having no subsequences of length 5 in which the first element is the largest.  Sergey Kitaev, Dec 11 2020
a(n) is the number of permutations p[1]..p[n] of {1,...,n} with p[j+1] < p[j]+4 for 0 < j < n.  Don Knuth, Apr 25 2022


REFERENCES

B. Polster, The Mathematics of Juggling, SpringerVerlag, 2003, p. 48.


LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..1662
Fan Chung and R. L. Graham, Primitive juggling sequences, Amer. Math. Monthly 115(3) (2008), 18519.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Kai Ting Keshia Yap, David Wehlau, and Imed Zaguia, Permutations Avoiding Certain Partiallyordered Patterns, arXiv:2101.12061 [math.CO], 2021.
Index entries for sequences related to juggling
Index entries for linear recurrences with constant coefficients, signature (4).


FORMULA

a(n) = n! for n <= 4, a(n) = 6*4^(n3) = A002023(n3) for n >= 3.
G.f.: 1 + x*(1  2*x  2*x^2)/(1  4*x).  Philippe Deléham, Aug 16 2005


MAPLE

A084509 := n > `if`((n<4), n!, 6*(4^(n3)));
INVERT([seq(A084519(n), n=1..12)]);


MATHEMATICA

LinearRecurrence[{4}, {1, 2, 6}, 30] (* Harvey P. Dale, Aug 23 2018 *)


CROSSREFS

First differences of A084508.
INVERT transform of A084519.
Cf. A002023, A084501, A084502, A084529, A145463.
Sequence in context: A179350 A179356 A179363 * A334767 A323260 A147915
Adjacent sequences: A084506 A084507 A084508 * A084510 A084511 A084512


KEYWORD

nonn,easy


AUTHOR

Antti Karttunen, Jun 02 2003


EXTENSIONS

a(0)=1 prepended by Alois P. Heinz, Dec 11 2020


STATUS

approved



