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

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

1,3

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

Table of n, a(n) for n=1..25.

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).

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

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 *)

Crossrefs

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

Sequence in context: A220117 A089930 A228529 * A304628 A265920 A262322

Adjacent sequences: A084516 A084517 A084518 * A084520 A084521 A084522

Keyword

nonn

Author

Antti Karttunen, Jun 02 2003

Status

approved