A208776 - OEIS

%I #26 Oct 31 2017 10:36:47

%S 7,13,19,36,63,143,299,719,1711,4249,10611,27144,69727,181467,475147,

%T 1253475,3324103,8862889,23729747,63791064,172066959,465577215,

%U 1263208683,3435919395,9366558151,25585896137,70019831931,191943278804,526978629663,1448862872667,3988658225035,10993823704779,30335737469495,83793424341677

%N Number of n-bead necklaces labeled with numbers 1..7 not allowing reversal, with no adjacent beads differing by more than 1.

%H Andrew Howroyd, <a href="/A208776/b208776.txt">Table of n, a(n) for n = 1..100</a>

%H 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 a(n) = Sum_{ d | n } A215338(d). - _Joerg Arndt_, Aug 13 2012

%F a(n) = (1/n) * Sum_{d | n} totient(n/d) * A124700(n). - _Andrew Howroyd_, Mar 18 2017

%e All solutions for n=3:

%e ..3....2....6....4....4....1....5....2....2....6....3....3....5....6....5....1

%e ..3....3....6....4....4....1....5....2....2....7....4....3....6....6....5....2

%e ..4....3....6....5....4....1....5....3....2....7....4....3....6....7....6....2

%e ..

%e ..4....7....1

%e ..5....7....1

%e ..5....7....2

%t sn[n_, k_] := 1/n*Sum[ DivisorSum[n, EulerPhi[#]*(1 + 2*Cos[i*Pi/(k + 1)])^(n/#) &], {i, 1, k}]; Table[sn[n, 7], {n, 1, 34}] // FullSimplify (* _Jean-François Alcover_, Oct 31 2017, after _Joerg Arndt_ *)

%o (PARI)

%o /* from the Knopfmacher et al. reference */

%o default(realprecision,99); /* using floats */

%o sn(n,k)=1/n*sum(i=1,k,sumdiv(n,j,eulerphi(j)*(1+2*cos(i*Pi/(k+1)))^(n/j)));

%o vector(66,n, round(sn(n,7)) )

%o /* _Joerg Arndt_, Aug 09 2012 */

%Y Column 7 of A208777.

%Y Cf. A215338 (cyclically smooth Lyndon words with 7 colors).

%K nonn

%O 1,1

%A _R. H. Hardin_, Mar 01 2012