A084521 -id:A084521 - cp's OEIS frontend

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.

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))));

A084523 Juggling states associated with the juggling sequence A084521.

Original entry on oeis.org

7, 7, 11, 7, 11, 13, 7, 19, 11, 7, 19, 13, 7, 11, 13, 14, 7, 11, 21, 14, 7, 19, 11, 13, 7, 19, 13, 14, 7, 19, 25, 13, 7, 19, 25, 14, 7, 35, 19, 11, 7, 35, 19, 13, 7, 35, 21, 11, 7, 35, 21, 14, 7, 35, 25, 13, 7, 35, 25, 14, 7, 11, 21, 26, 13, 7, 11, 37, 19, 13, 7, 11, 37, 26, 13, 7

Offset: 0

Antti Karttunen, Jun 02 2003

nonn

A. Karttunen, Scheme-program for computing this sequence

Same sequence in binary: A084524.

Cf. A084503, A084513.

a(0)=7, a(n) = (A084523(n-1) + 2^A084521(n) - 1)/2.

A084522 Successively larger 3-ball 'prime' ground-state site swaps of A084521 in concatenated decimal notation.

Original entry on oeis.org

3, 42, 441, 522, 531, 4440, 4530, 5241, 5340, 5511, 5520, 6222, 6231, 6312, 6330, 6411, 6420, 45501, 46131, 46401, 52440, 52530, 53502, 55140, 55500, 56112, 56130, 56202, 56400, 62241, 62340, 62511, 62520, 63141, 63501, 64140, 64500, 66111

Offset: 1

Antti Karttunen, Jun 02 2003

Note that this decimal representation works only up to the A084520(A084527(10))-1 = 1919th term which is 99600000, after which follows the 1920th 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 A084529. Subset of A084512.

A084526 The length of each successively larger, indecomposable, 'prime' 3-ball ground-state site swap given in A084521, i.e., the number of digits in each term of A084522.

Original entry on oeis.org

1, 2, 3, 3, 3, 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, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

Antti Karttunen, Jun 02 2003

nonn

A. Karttunen, Scheme-program for computing this sequence

Partial sums: A084525. Differs from A084516 first time at the 18th term, where A084516(18) = 4, while A084526(18) = 5. Cf. also A084506, A084520, A084528.

A084497 a(n) = n + A084521(n) - 3.

Original entry on oeis.org

1, 3, 2, 5, 6, 4, 9, 7, 8, 12, 11, 10, 14, 15, 16, 13, 18, 20, 19, 17, 23, 21, 24, 22, 27, 26, 28, 25, 31, 32, 29, 30, 35, 36, 34, 33, 40, 37, 38, 39, 44, 41, 43, 42, 48, 46, 45, 47, 52, 50, 51, 49, 56, 55, 53, 54, 60, 59, 58, 57, 62, 64, 65, 61, 63, 67, 70, 66, 69, 68, 72, 75

Offset: 1

Antti Karttunen, Jun 02 2003

nonn

Inverse: A084498. Cf. A084499, A084453, A084459, A084489, A084493.

A084527 First occurrence of n in A084521.

Original entry on oeis.org

16, 6, 3, 1, 2, 7, 37, 181, 799, 3503, 14365, 58033, 225215

Offset: 0

Antti Karttunen, Jun 02 2003

Antti Karttunen, Scheme-program for computing this sequence

Cf. A084507, A084517.

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.

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.

A084520 a(0)=0, after which each n occurs A084526(n) times.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 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, 18, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21

Offset: 0

Antti Karttunen, Jun 02 2003

nonn

Also minimum i such that A084525(i) >= n. Tells that the n-th throw (n>=1) in A084521 belongs to the a(n)-th lexicographical solution A084522(a(n)).

A. Karttunen, Scheme-program for computing this sequence

Differs from A084510 first time at the 65th term, where A084510(65) = 19, while A084520(65) = 18. Cf. A084500.

A084525 Partial sums of A084526.

Original entry on oeis.org

0, 1, 3, 6, 9, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 255, 260, 265

Offset: 0

Antti Karttunen, Jun 02 2003

nonn

For n>=1, (a(n-1)+1,a(n)) = (1,1),(2,3),(4,6),(7,9),(10,12),... gives the starting and ending offsets of the n-th site swap in the sequence A084521. Gives also the positions of 7s (ground states) in A084523.

A. Karttunen, Scheme-program for computing this sequence

Differs from A084515 first time at the 18th term, where A084515(18) = 64, while A084525(18) = 65. Cf. also A084505.

cp's OEIS Frontend

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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A084523 Juggling states associated with the juggling sequence A084521.

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A084522 Successively larger 3-ball 'prime' ground-state site swaps of A084521 in concatenated decimal notation.

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A084526 The length of each successively larger, indecomposable, 'prime' 3-ball ground-state site swap given in A084521, i.e., the number of digits in each term of A084522.

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A084497 a(n) = n + A084521(n) - 3.

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A084527 First occurrence of n in A084521.

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

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A084520 a(0)=0, after which each n occurs A084526(n) times.

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A084525 Partial sums of A084526.

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