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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. 4
 5, 9, 13, 24, 41, 91, 185, 435, 1009, 2445, 5945, 14813, 36977, 93465, 237313, 606465, 1556033, 4010205, 10367897, 26891385, 69930457, 182302161, 476262761, 1246710303, 3269321393, 8587489185, 22590646417, 59511087087, 156973954865, 414552479249, 1096017973385, 2900753690865, 7684758676201, 20377462520193 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..100 Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger, Smooth words and Chebyshev polynomials, arXiv:0809.0551v1 [math.CO], 2008. FORMULA a(n) = Sum_{ d | n } A215337(d). - Joerg Arndt, Aug 13 2012 a(n) = (1/n) * Sum_{d | n} totient(n/d) * A124698(n). - Andrew Howroyd, Mar 18 2017 EXAMPLE All solutions for n=3: ..5....1....3....2....2....4....4....4....1....3....2....3....1 ..5....1....4....2....2....4....5....4....1....3....3....3....2 ..5....1....4....3....2....5....5....4....2....3....3....4....2 MATHEMATICA 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 *) PROG (PARI) /* from the Knopfmacher et al. reference */ default(realprecision, 99); /* using floats */ sn(n, k)=1/n*sum(i=1, k, sumdiv(n, j, eulerphi(j)*(1+2*cos(i*Pi/(k+1)))^(n/j))); vector(66, n, round(sn(n, 5)) ) /* Joerg Arndt, Aug 09 2012 */ CROSSREFS Column 5 of A208777. Cf. A215337 (cyclically smooth Lyndon words with 5 colors). Sequence in context: A053411 A023382 A208718 * A271391 A151907 A151895 Adjacent sequences:  A208771 A208772 A208773 * A208775 A208776 A208777 KEYWORD nonn AUTHOR R. H. Hardin, Mar 01 2012 STATUS approved

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