A084501 -id:A084501 - cp's OEIS frontend

A084507 First occurrence of n in A084501.

Original entry on oeis.org

63, 17, 5, 1, 4, 18, 96, 510, 2580, 12522, 58920, 270870, 1223292

Offset: 0

Antti Karttunen, Jun 02 2003

Antti Karttunen, Scheme-program for computing this sequence

Cf. A084501, A084517, A084527.

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.

A084506 The length of each successively larger 3-ball ground-state site swap given in A084501, i.e., the number of digits in each term of A084502.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Antti Karttunen, Jun 02 2003

nonn

A. Karttunen, Table of n, a(n) for n = 1..32770
A. Karttunen, Scheme-program for computing this sequence
Index entries for sequences related to juggling

Partial sums: A084505.
Differs from A084556 first time at the 130th term, where A084506(130) = 6, while A084556(130) = 5.
Cf. A084516, A084526, A084500, A084508.

A084491 Permutation of Z, obtained by reflecting the juggling sequence A084501 from positive to negative numbers (with zero thrown at beat 0), folded to N with functions N2Z and Z2N.

Original entry on oeis.org

1, 8, 2, 10, 4, 12, 6, 16, 3, 14, 5, 18, 7, 20, 11, 22, 9, 24, 13, 28, 15, 26, 17, 32, 19, 30, 23, 34, 21, 38, 27, 40, 25, 36, 29, 46, 35, 42, 31, 44, 33, 52, 39, 50, 41, 48, 37, 54, 47, 56, 45, 58, 43, 60, 49, 62, 51, 64, 53, 68, 55, 66, 57, 70, 59, 74, 63, 72, 61, 76, 65, 78

Offset: 1

Antti Karttunen, Jun 02 2003

nonn

This permutation consists of three infinite cycles + infinite number of fixed points.

Inverse: A084492. Cf. also A084490, A084455, A084461, A084495, A084499, A065166.

[seq(Z2N(A084491_Z2Z(N2Z(n))),n=1..45)];
N2Z := n -> ((-1)^n)*floor(n/2);
Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);
A084491_Z2Z := z -> z+`if`((z > 0), A084501(z),`if`((z >= -3),2*(-z), A084501(A084490((-z)-3))));

A084503 Binary juggling state vectors (given in decimal), associated with the three balls juggling sequence A084501.

Original entry on oeis.org

7, 7, 7, 7, 11, 7, 7, 7, 7, 7, 11, 7, 11, 7, 7, 11, 13, 7, 19, 11, 7, 19, 13, 7, 7, 7, 7, 7, 7, 7, 11, 7, 7, 11, 7, 7, 7, 11, 13, 7, 7, 19, 11, 7, 7, 19, 13, 7, 11, 7, 7, 7, 11, 7, 11, 7, 11, 13, 7, 7, 11, 13, 14, 7, 11, 21, 11, 7, 11, 21, 14, 7, 19, 11, 7, 7, 19, 11, 13, 7, 19, 13, 7, 7, 19

Offset: 0

Antti Karttunen, Jun 02 2003

nonn

A. Karttunen, Table of n, a(n) for n = 0..283991
A. Karttunen, Scheme-program for computing this sequence
Index entries for sequences related to juggling

Same sequence in binary: A084504.
Cf. A084513, A084523.
A084468 gives the records.

a(0)=7, a(n)=(A084503(n-1)+(2^A084501(n))-1)/2.

A084489 a(n) = n + A084501(n) - 3.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 7, 8, 9, 11, 10, 13, 12, 14, 16, 17, 15, 20, 18, 19, 23, 22, 21, 24, 25, 26, 27, 28, 29, 31, 30, 32, 34, 33, 35, 36, 38, 39, 37, 40, 43, 41, 42, 44, 47, 46, 45, 49, 48, 50, 51, 53, 52, 55, 54, 57, 58, 56, 59, 61, 62, 63, 60, 65, 67, 64, 66, 69, 71, 70, 68, 74

Offset: 1

Antti Karttunen, Jun 02 2003

nonn

Inverse: A084490. Cf. A084491, A084453, A084459, A084493, A084497.

A084502 Successively larger 3-ball ground-state site swaps of A084501 in concatenated decimal notation.

Original entry on oeis.org

3, 33, 42, 333, 342, 423, 441, 522, 531, 3333, 3342, 3423, 3441, 3522, 3531, 4233, 4242, 4413, 4440, 4512, 4530, 5223, 5241, 5313, 5340, 5511, 5520, 6222, 6231, 6312, 6330, 6411, 6420, 33333, 33342, 33423, 33441, 33522, 33531, 34233, 34242, 34413

Offset: 1

Antti Karttunen, Jun 02 2003

Note that this decimal representation works only up to the A084500(A084507(10))-1 = 7707th term which is 99600000, after which follows the 7708th 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 'digits' in term a(n) is given by A084506.
The number of terms of length n is given by A084509.
Cf. A084512, A084522.

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

A084500 a(0)=0, after which each n occurs A084506(n) times.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23

Offset: 0

Antti Karttunen, Jun 02 2003

nonn

Also minimum i such that A084505(i) >= n. Tells that the n-th throw (n>=1) in A084501 belongs to the a(n)-th lexicographical solution A084502(a(n)).

Differs from A084557 first time at the 605th term, where A084500(605) = 130, while A084557(605) = 131. Cf. A084510, A084520.

A084505 Partial sums of A084506.

Original entry on oeis.org

0, 1, 3, 5, 8, 11, 14, 17, 20, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 83, 87, 91, 95, 99, 103, 107, 111, 115, 119, 124, 129, 134, 139, 144, 149, 154, 159, 164, 169, 174, 179, 184, 189, 194, 199, 204, 209, 214, 219, 224, 229, 234, 239, 244, 249

Offset: 0

Antti Karttunen, Jun 02 2003

nonn

For n>=1, (a(n-1)+1,a(n)) = (1,1),(2,3),(4,5),(6,8),(9,11),(12,14),... gives the starting and ending offsets of the n-th site swap in the sequence A084501.

Note: this sequence is related to juggling with three objects. For a similar sequence related to (ordinary) permutations, see A084555.

A. Karttunen, Table of n, a(n) for n = 0..32770
A. Karttunen, Scheme-program for computing this sequence
Index entries for sequences related to juggling

Differs from A084555 first time at the 130th term, where A084505(130) = 605, while A084555(130) = 604.

A084503(a(n)) = 7 for all n. However, this does not give the positions of all 7s in A084503 because the state sequences are not all indecomposable. Cf. A084515, A084525.

Added the b-file, moved the misplaced Mathematica code to A084555, where it belongs. - Antti Karttunen, Oct 24 2012

cp's OEIS Frontend

A084507 First occurrence of n in A084501.

Views

Author

Keywords

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A084506 The length of each successively larger 3-ball ground-state site swap given in A084501, i.e., the number of digits in each term of A084502.

Views

Author

Keywords

Links

Crossrefs

A084491 Permutation of Z, obtained by reflecting the juggling sequence A084501 from positive to negative numbers (with zero thrown at beat 0), folded to N with functions N2Z and Z2N.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A084503 Binary juggling state vectors (given in decimal), associated with the three balls juggling sequence A084501.

Views

Author

Keywords

Links

Crossrefs

Formula

A084489 a(n) = n + A084501(n) - 3.

Views

Author

Keywords

Links

Crossrefs

A084502 Successively larger 3-ball ground-state site swaps of A084501 in concatenated decimal notation.

Views

Author

Keywords

Comments

Links

Crossrefs

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A084500 a(0)=0, after which each n occurs A084506(n) times.

Views

Author

Keywords

Comments

Links

Crossrefs

A084505 Partial sums of A084506.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions