A032169 - OEIS

%I #64 Feb 23 2025 09:58:33

%S 1,6,26,91,273,728,1768,3978,8398,16796,32065,58786,104006,178296,

%T 297160,482885,766935,1193010,1820910,2731365,4032015,5864749,8414640,

%U 11920740,16689036,23107896,31666376,42975796

%N Number of aperiodic necklaces of n beads of 2 colors, 11 of them black.

%C From _Petros Hadjicostas_, Aug 26 2018: (Start)

%C Assume n >= k >= 2. If a_k(n) is the number of aperiodic necklaces of n beads of 2 colors such that k of them are black and n-k of them are white, then a_k(n) = (1/k)*Sum_{d|gcd(n,k)} mu(d)*binomial(n/d - 1, k/d - 1) = (1/n)*Sum_{d|gcd(n,k)} mu(d)*binomial(n/d, k/d). This follows from Herbert Kociemba's general formula for the g.f. of (a_k(n): n>=1) that can be found in the comments for sequence A032168.

%C For k prime, we get a_k(n) = floor(binomial(n-1, k-1)/k). In such a case, the sequence becomes a column for triangle A011847. (This is not true when k is composite >= 4.)

%C (End)

%H Ray Chandler, <a href="/A032169/b032169.txt">Table of n, a(n) for n = 12..1012</a>

%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 <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>

%H <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1,1,-10,45,-120,210,-252,210,-120,45,-10,1).

%F "CHK[ 11 ]" (necklace, identity, unlabeled, 11 parts) transform of 1, 1, 1, 1, ...

%F G.f.: (x^11/11)*(1/(1-x)^11-1/(1-x^11)). - _Herbert Kociemba_, Oct 16 2016

%F a(n) = (1/11)*(binomial(n-1, 10) - I(11|n)) = floor(binomial(n-1, 10)/11) for n >= 12, where I(a|b) = 1 if integer a divides integer b, and 0 otherwise. - _Petros Hadjicostas_, Aug 26 2018

%t CoefficientList[Series[x^11/11 (1/(1-x)^11-1/(1- x^11)),{x,0,50}],x] (* _Herbert Kociemba_, Oct 16 2016 *)

%Y A column of triangle A011847.

%K nonn,changed

%O 12,2

%A _Christian G. Bower_