A208773 - OEIS

%I #24 Oct 31 2017 10:36:26

%S 4,7,10,18,30,65,128,293,658,1544,3622,8711,20924,50889,124150,304718,

%T 750334,1855429,4600696,11442853,28528618,71294416,178529670,

%U 447923761,1125756860,2833917147,7144466842,18036449390,45591671454,115381885423,292329164912,741411257693,1882219950046,4782783122992,12163730636250

%N Number of n-bead necklaces labeled with numbers 1..4 not allowing reversal, with no adjacent beads differing by more than 1.

%H Andrew Howroyd, <a href="/A208773/b208773.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 } A215336(d). - _Joerg Arndt_, Aug 13 2012

%F a(n) = (1/n) * Sum_{d | n} totient(n/d) * A124697(n). - _Andrew Howroyd_, Mar 18 2017

%e All solutions for n=3:

%e ..2....4....1....2....1....2....3....3....1....3

%e ..2....4....1....2....1....3....3....4....2....3

%e ..3....4....2....2....1....3....3....4....2....4

%t 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, 4], {n, 1, 35}] // 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,4)) )

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

%Y Column 4 of A208777.

%Y Cf. A215336 (cyclically smooth Lyndon words with 4 colors).

%K nonn

%O 1,1

%A _R. H. Hardin_, Mar 01 2012