A084507 -id:A084507 - cp's OEIS frontend

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.

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.

A084501 An infinite juggling sequence of three balls: successively larger ground-state 3-ball site swaps listed in lexicographic order.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 02 2003

Every possible 3-ball asynchronic site swap of finite period occurs as a subsequence of this sequence. E.g., "51" (three-ball shower) occurs first time at a(65)=5, a(66)=1.
We obtain the sequence by traversing each possible loop of successively larger lengths in 3-ball state graph as depicted in Polster's book, or section 7 of Knutson's Siteswap FAQ (but not limited by throw height), starting from and ending to the ground state 7 (xxx) and by concatenating those sequences in lexicographic order.
One can take any subsequence A084501[i..j] such that A084503(i-1) = A084503(j) = 7 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 first 39 terms, one gets a site swap pattern "333423333424234415225313333334234233441".

			The successive site swaps are: 3; 3,3; 4,2; 3,3,3; 3,4,2; 4,2,3; 4,4,1; 5,2,2; 5,3,1; 3,3,3,3; ... See A084502.

B. Polster, The Mathematics of Juggling, Springer-Verlag, 2003, p. 45.

A. Karttunen, Table of n, a(n) for n = 1..283991
A. Karttunen, Scheme-program for computing this sequence
A. Knutson, Siteswap FAQ, Section 7, A deeper level: landing schedules, or juggling states
Juggling Information Service, Site Swap animators and other juggling software
Index entries for sequences related to juggling

Subsets: A084511, A084521.
The number of such site swaps of length n is given by A084509.
First position where n appears: A084507.

A084521 An infinite juggling sequence of three balls: successively larger 'prime' ground-state 3-ball site swaps listed in lexicographical order. A subset of A084511.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 02 2003

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.

			The successive site swaps are: 3; 4,2; 4,4,1; 5,2,2; 5,3,1; 4,4,4,0; 4,5,3,0; ... See A084522.

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

The number of such site swaps of length n is given by A084529. First position where n appears: A084527.

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.

A084455 Permutation of Z, obtained by reflecting the juggling sequence A084452 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, 10, 2, 12, 4, 14, 6, 8, 9, 20, 3, 22, 5, 24, 7, 16, 17, 18, 19, 32, 11, 28, 13, 34, 15, 26, 27, 38, 23, 30, 31, 44, 21, 46, 25, 36, 37, 50, 29, 40, 41, 42, 43, 58, 33, 52, 35, 48, 49, 62, 39, 64, 47, 54, 55, 56, 57, 72, 45, 60, 61, 74, 51, 76, 53, 66, 67, 68, 69, 70, 71, 86

Offset: 1

Antti Karttunen, Jun 02 2003

nonn

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

Inverse: A084466. Cf. also A084454, A084461, A084491, A084495, A084499, A065166.

[seq(Z2N(A084455_Z2Z(N2Z(n))),n=1..45)];
N2Z := n -> ((-1)^n)*floor(n/2);
Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);
A084455_Z2Z := z -> z+`if`((z > 0), A084452(z),`if`((z >= -3),2*(-z), A084452(A084454((-z)-3))));

A084461 Permutation of Z, obtained by reflecting the juggling sequence A084458 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, 10, 2, 12, 4, 18, 6, 8, 9, 20, 3, 28, 5, 14, 15, 16, 17, 24, 7, 34, 11, 22, 23, 38, 19, 26, 27, 46, 13, 30, 31, 32, 33, 48, 21, 36, 37, 58, 25, 40, 41, 42, 43, 44, 45, 60, 29, 72, 35, 50, 51, 52, 53, 54, 55, 56, 57, 66, 39, 80, 47, 62, 63, 64, 65, 84, 59, 68, 69, 70, 71, 94

Offset: 1

Antti Karttunen, Jun 02 2003

nonn

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

Inverse: A084462. Cf. also A084460, A084455, A084491, A084495, A084499, A065166.

[seq(Z2N(A084461_Z2Z(N2Z(n))),n=1..45)];
N2Z := n -> ((-1)^n)*floor(n/2);
Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);
A084461_Z2Z := z -> z+`if`((z > 0), A084458(z),`if`((z >= -3),2*(-z), A084458(A084460((-z)-3))));

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))));

A084495 Permutation of Z, obtained by reflecting the juggling sequence A084511 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, 12, 4, 10, 6, 16, 3, 18, 7, 14, 5, 24, 13, 20, 9, 22, 11, 30, 17, 28, 19, 26, 15, 34, 25, 36, 23, 38, 21, 32, 33, 42, 27, 46, 29, 40, 31, 44, 39, 50, 35, 54, 41, 52, 37, 48, 49, 60, 43, 56, 47, 62, 45, 58, 53, 68, 57, 66, 51, 70, 55, 64, 65, 76, 61, 78, 59, 72, 63, 74

Offset: 1

Antti Karttunen, Jun 02 2003

nonn

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

Inverse: A084496. Cf. also A084494, A084455, A084461, A084491, A084499, A065166.

[seq(Z2N(A084495_Z2Z(N2Z(n))),n=1..45)];
N2Z := n -> ((-1)^n)*floor(n/2);
Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);
A084495_Z2Z := z -> z+`if`((z > 0), A084511(z),`if`((z >= -3),2*(-z), A084511(A084494((-z)-3))));

A084499 Permutation of Z, obtained by reflecting the juggling sequence A084521 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, 12, 4, 10, 6, 16, 3, 18, 7, 14, 5, 24, 13, 20, 9, 22, 11, 30, 17, 28, 19, 26, 15, 34, 25, 36, 23, 38, 21, 32, 33, 42, 27, 46, 29, 44, 31, 40, 41, 52, 35, 48, 39, 54, 37, 50, 45, 60, 49, 58, 43, 62, 47, 56, 57, 68, 53, 70, 51, 64, 55, 66, 63, 76, 65, 78, 59, 74, 61, 72

Offset: 1

Antti Karttunen, Jun 02 2003

nonn

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

Inverse: A084530. Cf. also A084498, A084455, A084461, A084491, A084495, A065166.

[seq(Z2N(A084499_Z2Z(N2Z(n))),n=1..45)];
N2Z := n -> ((-1)^n)*floor(n/2);
Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);
A084499_Z2Z := z -> z+`if`((z > 0), A084511(z),`if`((z >= -3),2*(-z), A084521(A084498((-z)-3))));

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.

cp's OEIS Frontend

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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A084501 An infinite juggling sequence of three balls: successively larger ground-state 3-ball site swaps listed in lexicographic order.

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A084521 An infinite juggling sequence of three balls: successively larger 'prime' ground-state 3-ball site swaps listed in lexicographical order. A subset of A084511.

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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.

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A084455 Permutation of Z, obtained by reflecting the juggling sequence A084452 from positive to negative numbers (with zero thrown at beat 0), folded to N with functions N2Z and Z2N.

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Maple

A084461 Permutation of Z, obtained by reflecting the juggling sequence A084458 from positive to negative numbers (with zero thrown at beat 0), folded to N with functions N2Z and Z2N.

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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.

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Maple

A084495 Permutation of Z, obtained by reflecting the juggling sequence A084511 from positive to negative numbers (with zero thrown at beat 0), folded to N with functions N2Z and Z2N.

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Maple

A084499 Permutation of Z, obtained by reflecting the juggling sequence A084521 from positive to negative numbers (with zero thrown at beat 0), folded to N with functions N2Z and Z2N.

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Maple

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

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