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A084519 Number of indecomposable ground-state 3-ball juggling sequences of period n. 7
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 (list; graph; refs; listen; history; text; internal format)
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 * A265920 A262322 A180278

Adjacent sequences:  A084516 A084517 A084518 * A084520 A084521 A084522

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 02 2003

STATUS

approved

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Last modified December 11 19:09 EST 2017. Contains 295919 sequences.