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A084509
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Number of ground-state 3-ball juggling sequences of period n.
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10
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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
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0,3
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COMMENTS
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This sequence counts the length n asynchronic site swaps given in A084501/A084502.
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
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REFERENCES
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B. Polster, The Mathematics of Juggling, Springer-Verlag, 2003, p. 48.
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LINKS
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FORMULA
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a(n) = n! for n <= 4, a(n) = 6*4^(n-3) = A002023(n-3) for n >= 3.
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MAPLE
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A084509 := n -> `if`((n<4), n!, 6*(4^(n-3)));
INVERT([seq(A084519(n), n=1..12)]);
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MATHEMATICA
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LinearRecurrence[{4}, {1, 2, 6}, 30] (* Harvey P. Dale, Aug 23 2018 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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