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A110710 Number of ternary necklaces with n beads of each color and no adjacent beads of the same color (i.e., no substrings 00, 11, 22). 8
1, 2, 5, 16, 70, 348, 1948, 11444, 70380, 445944, 2896590, 19186740, 129186596, 881808728, 6089851874, 42482906040, 298976142764, 2120377458900, 15141289233972, 108784152585236, 785869931659980, 5705406374249272 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The number of circular arrangements (counted up to rotations) of n blue, n red and n green items such that there are no adjacent items of the same color. The number of various linear arrangements is given by A110706, A110707 and A110711.
LINKS
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
FORMULA
a(n) = Sum_{d|n} A110707(n/d)*eulerphi(d) / (3n) for n>0, a(0)=1.
a(n) ~ sqrt(3) * 2^(3*n - 1) / (Pi * n^2). - Vaclav Kotesovec, Mar 20 2023
EXAMPLE
For n=2 there are 5 necklaces: 010212, 012012, 012021, 012102, 021021.
MATHEMATICA
b = Binomial; A110707[n_] := 2*Sum[b[n - 1, k]*(b[n - 1, k]*(b[2*n + 1 - 2*k, n + 1] - 3*b[2*n - 1 - 2*k, n + 1]) + b[n - 1, k + 1]*(b[2*n - 2*k, n + 1] - 3*b[2*n - 2*k - 2, n + 1])), {k, 0, n/2}]; a[n_] := DivisorSum[n, A110707[n/#]*EulerPhi[#]&]/(3n); a[0]=1; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Dec 04 2015, adapted from PARI *)
PROG
(PARI) { A110707(n) = 2 * sum(k=0, n\2, binomial(n-1, k) * (binomial(n-1, k)*(binomial(2*n+1-2*k, n+1)-3*binomial(2*n-1-2*k, n+1)) + binomial(n-1, k+1)*(binomial(2*n-2*k, n+1)-3*binomial(2*n-2*k-2, n+1)) )); A110710(n) = sumdiv(n, d, A110707(n\d)*eulerphi(d))\(3*n); }
CROSSREFS
Sequence in context: A259408 A251684 A129092 * A245881 A078639 A002632
KEYWORD
nonn
AUTHOR
Max Alekseyev, Aug 05 2005
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Dec 04 2015
STATUS
approved

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)