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A032294 Number of aperiodic bracelets (turnover necklaces) with n beads of 3 colors. 1

%I #33 Apr 30 2019 22:08:04

%S 3,3,7,15,36,79,195,477,1209,3168,8415,22806,62412,172887,481552,

%T 1351485,3808080,10780653,30615351,87226932,249144506,713378655,

%U 2046856563,5884468110,16946569332,48883597728,141217159239

%N Number of aperiodic bracelets (turnover necklaces) with n beads of 3 colors.

%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>

%H F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>

%H F. Ruskey, <a href="/A000011/a000011.pdf">Necklaces, Lyndon words, De Bruijn sequences, etc.</a> [Cached copy, with permission, pdf format only]

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>

%F MOEBIUS transform of A027671.

%F From _Herbert Kociemba_, Nov 28 2016: (Start)

%F More generally, gf(k) is the g.f. for the number of bracelets with primitive period n and beads of k colors.

%F gf(k): Sum_{n>=1} mu(n)*( -log(1-k*x^n)/n + Sum_{i=0..2} binomial(k,i)x^(n*i)/(1-k*x^(2*n)) )/2. (End)

%t mx=40;gf[x_,k_]:=Sum[ MoebiusMu[n]*(-Log[1-k*x^n]/n+Sum[Binomial[k,i]x^(n i),{i,0,2}]/( 1-k x^(2n)))/2,{n,mx}]; CoefficientList[Series[gf[x,3],{x,0,mx}],x] (* _Herbert Kociemba_, Nov 28 2016 *)

%o (PARI) a(x, k) = sum(n=1, 40, moebius(n) * (-log(1 - k*x^n )/n + sum(i=0, 2, binomial(k, i) * x^(n*i)) / (1 - k* x^(2*n)))/2);

%o Vec(a(x, 3) + O(x^41)) \\ _Indranil Ghosh_, Mar 29 2017

%Y Column 3 of A276550.

%K nonn

%O 1,1

%A _Christian G. Bower_

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