A208775 Number of n-bead necklaces labeled with numbers 1..6 not allowing reversal, with no adjacent beads differing by more than 1.
6, 11, 16, 30, 52, 117, 242, 577, 1360, 3347, 8278, 20978, 53346, 137422, 355978, 928731, 2434580, 6414014, 16961468, 45017417, 119840582, 319916277, 856089572, 2295950281, 6169664562, 16608996492, 44785220118, 120942143132, 327053057574, 885545659155, 2400570958904, 6514679288762, 17697582670400, 48122529680805
Offset: 1
Keywords
Examples
All solutions for n=3: ..5....1....1....3....5....5....1....2....2....3....3....6....2....4....4....4 ..5....1....2....3....6....5....1....3....2....3....4....6....2....4....5....4 ..5....2....2....4....6....6....1....3....2....3....4....6....3....4....5....5
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:0809.0551 [math.CO], 2008.
Crossrefs
Column 6 of A208777.
Programs
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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, 6], {n, 1, 34}] // FullSimplify (* Jean-François Alcover, Oct 31 2017, after Joerg Arndt *) -
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,6)) ) /* Joerg Arndt, Aug 09 2012 */
Formula
a(n) = (1/n) * Sum_{d | n} totient(n/d) * A124699(n). - Andrew Howroyd, Mar 18 2017