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

Author

Antti Karttunen, Jun 02 2003

Keywords

• nonn

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

• 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).

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