A065178 Number of site swap patterns with 2 balls and exact period n.

1, 2, 6, 15, 42, 107, 294, 780, 2128, 5781, 15918, 43885, 122010, 340323, 954394, 2685930, 7588770, 21507696, 61144062, 174283887, 498012094, 1426213191, 4092816966, 11767176070, 33890202192, 97761428205, 282424564744

Offset: 1

Author

Antti Karttunen, Oct 19 2001

Keywords

• nonn

Comments

When interspersed with 0's, exponents in expansion of A065481 as a product zeta(n)^(-a(n)).

Examples

			We have one period 1 (2), two period 2 (31/13 and 40/04) and six period three 2-ball siteswaps (312, 330, 411, 420, 501, 600) (The average of the digits is always 2).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Juggling Information Service, Site Swap FAQs
• M. Macauley, Braids and Juggling patterns, Thesis (Harvey Mudd College) 2003, eq. (A1).
• G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

Crossrefs

Row 2 of A065177.

Cf. A008683, A065179, A065180, A065481.

Programs

Maple

[seq(DistSS(p,2),p=1..60)];
A065178 := proc(n)
    add( mobius(n/d)*(3^d-2^d),d=numtheory[divisors](n)) /n ;
end proc:
seq(A065178(n),n=1..30) ; # R. J. Mathar, Aug 05 2015

Mathematica

a[n_] := DivisorSum[n, MoebiusMu[n/#] * (3^#-2^#)&] / n; Array[a, 30] (* Jean-François Alcover, Mar 05 2016, after R. J. Mathar *)

Formula

a(n) ~ 3^n/n. - Vaclav Kotesovec, Mar 05 2016

Inverse Euler transform of A133494. - Alois P. Heinz, Jun 23 2018

G.f.: Sum_{k>=1} mu(k) * log(1 + x^k/(1 - 3*x^k))/k. - Seiichi Manyama, Apr 14 2025