A084519 -id:A084519 - cp's OEIS frontend

A084518 Partial sums of A084519. Positions of ones in the first differences of A084516.

1, 2, 5, 18, 65, 238, 877, 3234, 11929, 44006, 162341, 598890, 2209361, 8150542, 30068125, 110924178, 409209865, 1509614198, 5569110677, 20544980154, 75792390209, 279605352286, 1031490797581, 3805267877730, 14037995932921

Offset: 1

Antti Karttunen, Jun 02 2003

nonn

Index entries for linear recurrences with constant coefficients, signature (3, 2, 2).

Accumulate[LinearRecurrence[{3,2,2},{1,1,3},30]] (* Harvey P. Dale, Jul 20 2013 *)

Empirical G.f.: x*(1-2*x-2*x^2)/((1-x)*(1-3*x-2*x^2-2*x^3)). [Colin Barker, Apr 17 2012]

A084511 An infinite juggling sequence of three balls: successively larger indecomposable ground-state 3-ball site swaps listed in lexicographical order. A subset of A084501.

Original entry on oeis.org

3, 4, 2, 4, 4, 1, 5, 2, 2, 5, 3, 1, 4, 4, 4, 0, 4, 5, 1, 2, 4, 5, 3, 0, 5, 2, 4, 1, 5, 3, 4, 0, 5, 5, 1, 1, 5, 5, 2, 0, 6, 2, 2, 2, 6, 2, 3, 1, 6, 3, 1, 2, 6, 3, 3, 0, 6, 4, 1, 1, 6, 4, 2, 0, 4, 4, 5, 0, 2, 4, 5, 1, 4, 1, 4, 5, 5, 0, 1, 4, 6, 1, 2, 2, 4, 6, 1, 3, 1, 4, 6, 3, 0, 2, 4, 6, 4, 0, 1, 5, 2, 4, 4, 0, 5

Offset: 1

Antti Karttunen, Jun 02 2003

By "indecomposable" we mean that the juggling state sequence associated to each loop should not return to the ground state 7 (xxx) until after the last throw. I.e., this means that A084515 gives positions of ALL the 7s (ground states) in A084513.

One can take any subsequence A084511[A084515(i)+1..A084515(j)] (j>i) and try to juggle it periodically or give it to one of the Siteswap animators available at J.I.S., e.g., by taking the terms 4-12, one gets a site swap pattern "441522531".

			The successive site swaps are: 3; 4,2; 4,4,1; 5,2,2; 5,3,1; 4,4,4,0; 4,5,1,2; 4,5,3,0; ... See A084512.

Juggling Information Service, Site Swap animators and other juggling software
A. Karttunen, Scheme-program for computing this sequence
Index entries for sequences related to juggling

Subset: A084521.
The number of such site swaps of length n is given by A084519.
First position where n appears: A084517.

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

Original entry on oeis.org

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

Antti Karttunen, Jun 02 2003

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

B. Polster, The Mathematics of Juggling, Springer-Verlag, 2003, p. 48.

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).

First differences of A084508.
INVERT transform of A084519.
Cf. A002023, A084501, A084502, A084529, A145463.

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

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

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

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

A084512 Successively larger 3-ball indecomposable ground-state site swaps of A084511 in concatenated decimal notation.

Original entry on oeis.org

3, 42, 441, 522, 531, 4440, 4512, 4530, 5241, 5340, 5511, 5520, 6222, 6231, 6312, 6330, 6411, 6420, 44502, 45141, 45501, 46122, 46131, 46302, 46401, 52440, 52512, 52530, 53502, 55140, 55500, 56112, 56130, 56202, 56400, 62241, 62340, 62511

Offset: 1

Antti Karttunen, Jun 02 2003

Note that this decimal representation works only up to the A084510(A084517(10))-1 = 2748th term which is 99600000, after which follows the 2749th solution 10,2,2,2,2,2,2,2 which would be usually represented as "A2222222".

A. Karttunen, Scheme-program for computing this sequence
Index entries for sequences related to juggling

The number of terms of length n is given by A084519.

Subset of A084502. Cf. A084522.

A084529 Number of 'prime' ground-state 3-ball juggling sequences of period n.

Original entry on oeis.org

1, 1, 3, 12, 42, 142, 502, 1702, 5878

Offset: 1

Antti Karttunen, Jun 02 2003

nonn

This sequence counts the length n asynchronic site swaps given in A084521/A084522.

A juggling sequence is defined as 'prime', if it does not visit any state more than once. This means that in A084523 no integer occurs twice between consecutive sevens.

B. Polster, The Mathematics of Juggling, Springer-Verlag, 2003, pp. 50-51.

A. Karttunen, Scheme-program for counting this sequence
Index entries for sequences related to juggling

First differences of A084528. Cf. A084509, A084519.

A145463 Eigentriangle, row sums = A084509.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 13, 3, 2, 6, 47, 13, 6, 6, 24, 173, 47, 26, 18, 24, 96, 639, 173, 94, 78, 72, 96, 384, 2357, 639, 346, 282, 312, 288, 384, 1536, 8695, 2357, 1278, 1038, 1128, 1248, 1152, 1536, 6144, 32077, 8695, 4714, 3834, 4152, 4512, 4992, 4608, 6144, 24576

Offset: 1

Gary W. Adamson, Oct 11 2008

Row sums = A084509: (1, 2, 6, 24, 96, 384, 1536,...).
Right border = A084509 shifted: (1, 1, 2, 6, 24,...).
Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
1;
1, 1;
3, 1, 2;
13, 3, 2, 6;
47, 13, 6, 6, 24;
173, 47, 26, 18, 24, 96;
639, 173, 94, 78, 72, 96, 384;
2357, 639, 346, 282, 312, 288, 384, 1536;
...
Row 4 = (13, 3, 2, 6) = termwise products of (13, 3, 1, 1) and (1, 1, 2, 6).

A084509, Cf. A084519

Triangle read by rows, M * (A084509 * 0^(n-k)). M = an infinite lower triangular matrix with A084519: (1, 1, 3, 13, 47, 173,...) in every column; and (A084509 * 0^(n-k)) = an infinite lower triangular matrix with A084509 (1, 2, 6, 24, 96,...) shifted: (1, 1, 2, 6, 24, 96, 384,...) as the right diagonal and the rest zeros.

cp's OEIS Frontend

A084518 Partial sums of A084519. Positions of ones in the first differences of A084516.

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A084511 An infinite juggling sequence of three balls: successively larger indecomposable ground-state 3-ball site swaps listed in lexicographical order. A subset of A084501.

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

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A084512 Successively larger 3-ball indecomposable ground-state site swaps of A084511 in concatenated decimal notation.

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A084529 Number of 'prime' ground-state 3-ball juggling sequences of period n.

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A145463 Eigentriangle, row sums = A084509.

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