cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

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Author

Antti Karttunen, Jun 02 2003

Keywords

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Crossrefs

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.

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

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Author

Antti Karttunen, Jun 02 2003

Keywords

Comments

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".

Examples

			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.

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Crossrefs

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

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

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Author

Antti Karttunen, Jun 02 2003

Keywords

Comments

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

References

  • 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

Links

Crossrefs

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

Programs

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

Formula

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

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

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Author

Antti Karttunen, Jun 02 2003

Keywords

Comments

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".

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The number of terms of length n is given by A084529. Subset of A084512.

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

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Author

Antti Karttunen, Jun 02 2003

Keywords

Comments

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".

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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.

A084510 a(0)=0, after which each n occurs A084516(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, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21
Offset: 0

Views

Author

Antti Karttunen, Jun 02 2003

Keywords

Comments

Also minimum i such that A084515(i) >= n. Tells that the n-th throw (n>=1) in A084511 belongs to the a(n)-th lexicographical solution A084512(a(n)).

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Crossrefs

Differs from A084520 first time at the 65th term, where A084510(65) = 19, while A084520(65) = 18.
Cf. A084500.
Showing 1-6 of 6 results.

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