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A208775 Number of n-bead necklaces labeled with numbers 1..6 not allowing reversal, with no adjacent beads differing by more than 1.

%I A208775 #20 Oct 31 2017 10:36:40
%S A208775 6,11,16,30,52,117,242,577,1360,3347,8278,20978,53346,137422,355978,
%T A208775 928731,2434580,6414014,16961468,45017417,119840582,319916277,
%U A208775 856089572,2295950281,6169664562,16608996492,44785220118,120942143132,327053057574,885545659155,2400570958904,6514679288762,17697582670400,48122529680805
%N A208775 Number of n-bead necklaces labeled with numbers 1..6 not allowing reversal, with no adjacent beads differing by more than 1.
%H A208775 Andrew Howroyd, <a href="/A208775/b208775.txt">Table of n, a(n) for n = 1..100</a>
%H A208775 Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger, <a href="http://arxiv.org/abs/0809.0551">Smooth words and Chebyshev polynomials</a>, arXiv:0809.0551 [math.CO], 2008.
%F A208775 a(n) = (1/n) * Sum_{d | n} totient(n/d) * A124699(n). - _Andrew Howroyd_, Mar 18 2017
%e A208775 All solutions for n=3:
%e A208775 ..5....1....1....3....5....5....1....2....2....3....3....6....2....4....4....4
%e A208775 ..5....1....2....3....6....5....1....3....2....3....4....6....2....4....5....4
%e A208775 ..5....2....2....4....6....6....1....3....2....3....4....6....3....4....5....5
%t A208775 sn[n_, k_] := 1/n*Sum[ Sum[ EulerPhi[j]*(1 + 2*Cos[i*Pi/(k + 1)])^(n/j), {j, Divisors[n]}], {i, 1, k}]; Table[sn[n, 6], {n, 1, 34}] // FullSimplify (* _Jean-François Alcover_, Oct 31 2017, after _Joerg Arndt_ *)
%o A208775 (PARI)
%o A208775 /* from the Knopfmacher et al. reference */
%o A208775 default(realprecision,99); /* using floats */
%o A208775 sn(n,k)=1/n*sum(i=1,k,sumdiv(n,j,eulerphi(j)*(1+2*cos(i*Pi/(k+1)))^(n/j)));
%o A208775 vector(66,n, round(sn(n,6)) )
%o A208775 /* _Joerg Arndt_, Aug 09 2012 */
%Y A208775 Column 6 of A208777.
%K A208775 nonn
%O A208775 1,1
%A A208775 _R. H. Hardin_, Mar 01 2012