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 1-5 of 5 results.

A084528 Partial sums of A084529. Positions of ones in the first differences of A084526.

Original entry on oeis.org

1, 2, 5, 17, 59, 201, 703, 2405
Offset: 1

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Author

Antti Karttunen, Jun 02 2003

Keywords

Links

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

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Author

Antti Karttunen, Jun 02 2003

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

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

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Author

Antti Karttunen, Jun 02 2003

Keywords

Comments

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

References

  • B. Polster, The Mathematics of Juggling, Springer-Verlag, 2003, p. 48.

Links

Crossrefs

First differences of A084508.
INVERT transform of A084519.
Cf. A002023, A084501, A084502, A084529, A145463.

Programs

  • Maple
    A084509 := n -> `if`((n<4),n!,6*(4^(n-3)));
INVERT([seq(A084519(n),n=1..12)]);
  • Mathematica
    LinearRecurrence[{4},{1,2,6},30] (* Harvey P. Dale, Aug 23 2018 *)

Formula

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

Extensions

a(0)=1 prepended by Alois P. Heinz, Dec 11 2020

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

Views

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

Views

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".

Links

Crossrefs

The number of terms of length n is given by A084529. Subset of A084512.
Showing 1-5 of 5 results.

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