OFFSET
1,1
COMMENTS
Allowing arbitrary differences between the first and last bead gives A215327. [Joerg Arndt, Aug 08 2012]
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..200
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 } A215335(d). - Joerg Arndt, Aug 13 2012
a(n) = (1/n) * Sum_{d | n} totient(n/d) * A124696(n). - Andrew Howroyd, Mar 18 2017
EXAMPLE
All solutions for n=4:
..1....2....2....2....1....1....1....3....2....1....2....1
..2....2....3....2....1....2....1....3....3....2....2....1
..1....3....2....2....2....3....1....3....3....2....2....1
..2....3....3....3....2....2....1....3....3....2....2....2
MATHEMATICA
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, 3], {n, 1, 36}] // FullSimplify (* 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, 3)) )
/* Joerg Arndt, Aug 09 2012 */
CROSSREFS
KEYWORD
nonn
AUTHOR
R. H. Hardin, Mar 01 2012
STATUS
approved