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A208774 Number of n-bead necklaces labeled with numbers 1..5 not allowing reversal, with no adjacent beads differing by more than 1.

%I A208774 #26 Oct 31 2017 10:36:32
%S A208774 5,9,13,24,41,91,185,435,1009,2445,5945,14813,36977,93465,237313,
%T A208774 606465,1556033,4010205,10367897,26891385,69930457,182302161,
%U A208774 476262761,1246710303,3269321393,8587489185,22590646417,59511087087,156973954865,414552479249,1096017973385,2900753690865,7684758676201,20377462520193
%N A208774 Number of n-bead necklaces labeled with numbers 1..5 not allowing reversal, with no adjacent beads differing by more than 1.
%H A208774 Andrew Howroyd, <a href="/A208774/b208774.txt">Table of n, a(n) for n = 1..100</a>
%H A208774 Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger, <a href="http://arxiv.org/abs/0809.0551">Smooth words and Chebyshev polynomials</a>, arXiv:0809.0551v1 [math.CO], 2008.
%F A208774 a(n) = Sum_{ d | n } A215337(d). - _Joerg Arndt_, Aug 13 2012
%F A208774 a(n) = (1/n) * Sum_{d | n} totient(n/d) * A124698(n). - _Andrew Howroyd_, Mar 18 2017
%e A208774 All solutions for n=3:
%e A208774 ..5....1....3....2....2....4....4....4....1....3....2....3....1
%e A208774 ..5....1....4....2....2....4....5....4....1....3....3....3....2
%e A208774 ..5....1....4....3....2....5....5....4....2....3....3....4....2
%t A208774 sn[n_, k_] := 1/n*Sum[DivisorSum[n, EulerPhi[#]*(1 + 2*Cos[i*Pi/(k + 1)])^(n/#)&], {i, 1, k}]; Table[sn[n, 5], {n, 1, 34}] // Simplify (* _Jean-François Alcover_, Oct 31 2017, after _Joerg Arndt_ *)
%o A208774 (PARI)
%o A208774 /* from the Knopfmacher et al. reference */
%o A208774 default(realprecision,99); /* using floats */
%o A208774 sn(n,k)=1/n*sum(i=1,k,sumdiv(n,j,eulerphi(j)*(1+2*cos(i*Pi/(k+1)))^(n/j)));
%o A208774 vector(66,n, round(sn(n,5)) )
%o A208774 /* _Joerg Arndt_, Aug 09 2012 */
%Y A208774 Column 5 of A208777.
%Y A208774 Cf. A215337 (cyclically smooth Lyndon words with 5 colors).
%K A208774 nonn
%O A208774 1,1
%A A208774 _R. H. Hardin_, Mar 01 2012