|
%I
%S 1,1,3,13,47,173,639,2357,8695,32077,118335,436549,1610471,5941181,
%T 21917583,80856053,298285687,1100404333,4059496479,14975869477,
%U 55247410055,203812962077,751885445295,2773777080149,10232728055191
%N Number of indecomposable ground-state 3-ball juggling sequences of period n.
%C This sequence counts the length n asynchronic site swaps given in A084511/A084512.
%C First differences of A084518. INVERTi transform of A084509. Cf. also A084529, A003319.
%C Equals left border of triangle A145463 [From _Gary W. Adamson_, Oct 11 2008]
%H Fan Chung, Ron Graham, <a href="http://www.jstor.org/stable/27642443">Primitive juggling sequences</a>, Am. Math. Monthly 115 (3) (2008) 185-194
%H <a href="/index/J#Juggling">Index entries for sequences related to juggling</a>
%F 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.]
%F G.f.: x*(1-2*x-2*x^2)/(1-3*x-2*x^2-2*x^3). [Colin Barker, Jan 14 2012]
%p INVERTi([seq(A084509(n),n=1..80)]);
%p 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;
%Y A145463 [From _Gary W. Adamson_, Oct 11 2008]
%K nonn
%O 1,3
%A _Antti Karttunen_ Jun 02 2003
|
