A208777 T(n,k) is the number of n-bead necklaces labeled with numbers 1..k not allowing reversal, with no adjacent beads differing by more than 1.
1, 2, 1, 3, 3, 1, 4, 5, 4, 1, 5, 7, 7, 6, 1, 6, 9, 10, 12, 8, 1, 7, 11, 13, 18, 19, 14, 1, 8, 13, 16, 24, 30, 39, 20, 1, 9, 15, 19, 30, 41, 65, 71, 36, 1, 10, 17, 22, 36, 52, 91, 128, 152, 60, 1, 11, 19, 25, 42, 63, 117, 185, 293, 315, 108, 1, 12, 21, 28, 48, 74, 143, 242, 435, 658, 685
Offset: 1
Examples
All solutions for n=4, k=3: ..2....1....2....1....2....2....2....1....3....1....1....1 ..3....2....2....2....2....3....2....1....3....1....2....1 ..2....2....3....1....2....3....2....1....3....2....3....1 ..3....2....3....2....2....3....3....2....3....2....2....1
Links
- R. H. Hardin, Table of n, a(n) for n = 1..456
- Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger, Smooth words and Chebyshev polynomials, arXiv:0809.0551v1 [math.CO], 2008.
Crossrefs
Programs
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Mathematica
T[n_, k_] := 1/n*Sum[DivisorSum[n, EulerPhi[#]*(1+2*Cos[i*Pi/(k+1)])^(n/#)&], {i, 1, k}] // FullSimplify; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 05 2015, adapted from PARI *) -
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))); T(n,k)=sn(n,k); matrix(22,22,n,k, round(T(n,k)) ) /* as matrix shown in comments */ /* Joerg Arndt, Aug 09 2012 */
Comments