A065179 Number of site swap patterns with 3 balls and exact period n.
1, 3, 12, 42, 156, 554, 2028, 7350, 26936, 98874, 365196, 1353520, 5039580, 18831306, 70626140, 265741350, 1002984060, 3796211692, 14406086604, 54801192684, 208932673508, 798218225802, 3055417434732, 11716355452900
Offset: 1
Keywords
Examples
We have one period 1 (3, the three-ball cascade), three period two (42/24, 51/15 = three-ball shower and 60/06) and twelve period three 3-ball siteswaps (423, 441, 450, 522, 531, 603, 612, 630, 711, 720, 801, 900) (The average of digits is always 3).
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, equation A1.
Programs
-
Maple
[seq(DistSS(p,3),p=1..60)]; A065179 := proc(n) add( mobius(n/d)*(4^d-3^d),d=numtheory[divisors](n)) /n ; end proc: seq(A065179(n),n=1..30) ; # R. J. Mathar, Aug 05 2015
-
Mathematica
a[n_] := DivisorSum[n, MoebiusMu[n/#]*(4^# - 3^#)&]/n; Array[a, 25] (* Jean-François Alcover, Mar 06 2016 *)
Formula
G.f.: Sum_{k>=1} mu(k) * log(1 + x^k/(1 - 4*x^k))/k. - Seiichi Manyama, Apr 14 2025