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

1,2

Comments

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

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]

Formula

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

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

Example

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).

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 *)

Crossrefs

Row 2 of A065177. Cf. A065179, A065180, A065481.

Sequence in context: A362566 A280782 A307308 * A178936 A221744 A338861

Adjacent sequences: A065175 A065176 A065177 * A065179 A065180 A065181

Keyword

nonn

Author

Antti Karttunen, Oct 19 2001

Status

approved