A084509 Number of ground-state 3-ball juggling sequences of period n.

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

Offset: 0

Author

Antti Karttunen, Jun 02 2003

Keywords

• nonn
• easy

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, Springer-Verlag, 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), 185-19.
• 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 Partially-ordered Patterns, arXiv:2101.12061 [math.CO], 2021.
• Index entries for sequences related to juggling
• Index entries for linear recurrences with constant coefficients, signature (4).

Crossrefs

First differences of A084508.

INVERT transform of A084519.

Cf. A002023, A084501, A084502, A084529, A145463.

Programs

Maple

A084509 := n -> `if`((n<4),n!,6*(4^(n-3)));
INVERT([seq(A084519(n),n=1..12)]);

Mathematica

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

Formula

a(n) = n! for n <= 4, a(n) = 6*4^(n-3) = A002023(n-3) for n >= 3.

G.f.: 1 + x*(1 - 2*x - 2*x^2)/(1 - 4*x). - Philippe Deléham, Aug 16 2005

Extensions

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