A084511 -id:A084511 - cp's OEIS frontend

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.

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.

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))));

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.

A084513 Juggling states associated with the juggling sequence A084511.

Original entry on oeis.org

7, 7, 11, 7, 11, 13, 7, 19, 11, 7, 19, 13, 7, 11, 13, 14, 7, 11, 21, 11, 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, 13, 22, 11, 7, 11, 21, 11, 13, 7, 11

Offset: 0

Antti Karttunen, Jun 02 2003

nonn

A. Karttunen, Scheme-program for computing this sequence

Same sequence in binary: A084514.

Cf. A084503, A084511, A084523.

a(0)=7, a(n) = (A084513(n-1) + 2^A084511(n) - 1)/2.

A084516 The length of each successively larger, indecomposable 3-ball ground-state site swap given in A084511, i.e., the number of digits in each term of A084512.

Original entry on oeis.org

1, 2, 3, 3, 3, 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, 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: A084515. Differs from A084526 first time at the 18th term, where A084516(18) = 4, while A084526(18) = 5. Cf. also A084506, A084510, A084518.

A084493 a(n) = n + A084511(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, 17, 19, 22, 24, 23, 21, 27, 25, 28, 26, 31, 30, 32, 29, 35, 36, 33, 34, 39, 40, 38, 37, 44, 41, 42, 43, 48, 45, 47, 46, 52, 50, 49, 51, 56, 54, 55, 53, 60, 59, 57, 58, 64, 63, 62, 61, 66, 67, 69, 65, 68, 71, 73, 70

Offset: 1

Antti Karttunen, Jun 02 2003

nonn

Inverse: A084494.

Cf. A084495, A084453, A084459, A084489, A084497.

A084517 First occurrence of n in A084511.

Original entry on oeis.org

16, 6, 3, 1, 2, 7, 41, 210, 1014, 4677, 20779, 89800, 379952

Offset: 0

Antti Karttunen, Jun 02 2003

Antti Karttunen, Scheme-program for computing this sequence

Cf. A084507, A084527.

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.

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))));

A084519 Number of indecomposable ground-state 3-ball juggling sequences of period n.

Original entry on oeis.org

1, 1, 3, 13, 47, 173, 639, 2357, 8695, 32077, 118335, 436549, 1610471, 5941181, 21917583, 80856053, 298285687, 1100404333, 4059496479, 14975869477, 55247410055, 203812962077, 751885445295, 2773777080149, 10232728055191

Offset: 1

Antti Karttunen, Jun 02 2003

nonn

This sequence counts the length n asynchronic site swaps given in A084511/A084512.
First differences of A084518. INVERTi transform of A084509. Cf. also A084529, A003319.
Equals left border of triangle A145463. - Gary W. Adamson, Oct 11 2008

Carsten Elsner, Dominic Klyve and Erik R. Tou, A zeta function for juggling sequences, Journal of Combinatorics and Number Theory, Volume 4, Issue 1, 2012, pp. 1-13; ISSN 1942-5600

Fan Chung, R. L. Graham, Primitive juggling sequences, Am. Math. Monthly 115 (3) (2008) 185-194
Index entries for sequences related to juggling
Index entries for linear recurrences with constant coefficients, signature (3, 2, 2).

Cf. A145463. - Gary W. Adamson, Oct 11 2008

INVERTi([seq(A084509(n),n=1..80)]);
with(combinat); A084519 := proc(n) option remember; local c,i,k; A084509(n)-add(add(mul(A084519(i),i=c),c=composition(n,k)),k=2..n); end;

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

a(n) seems to satisfy the recurrence: a(1) = a(2) = 1, a(3) = 3 and a(n) = 3*a(n-1)+2*a(n-2)+2*a(n-3). If so, a(n) = floor(A*B^n+1/2) where B = 3.6890953... is the real positive root of x^3-3x^2-2x-2 = 0 and A = 0.0687059... is the real positive root of 118*x^3+118*x^2+35*x-3 = 0. - Benoit Cloitre, Jun 14 2003 [This conjecture is established in the Chung-Graham paper.]

G.f.: x*(1-2*x-2*x^2)/(1-3*x-2*x^2-2*x^3). - Colin Barker, Jan 14 2012

cp's OEIS Frontend

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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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

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

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A084513 Juggling states associated with the juggling sequence A084511.

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A084516 The length of each successively larger, indecomposable 3-ball ground-state site swap given in A084511, i.e., the number of digits in each term of A084512.

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A084493 a(n) = n + A084511(n) - 3.

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A084517 First occurrence of n in A084511.

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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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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

A084519 Number of indecomposable ground-state 3-ball juggling sequences of period n.

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