

A208777


T(n,k) is the number of nbead necklaces labeled with numbers 1..k not allowing reversal, with no adjacent beads differing by more than 1.


8



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
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OFFSET

1,2


COMMENTS

Table starts
.1..2...3...4...5...6...7...8....9...10...11...12...13...14...15...16...17...18
.1..3...5...7...9..11..13..15...17...19...21...23...25...27...29...31...33...35
.1..4...7..10..13..16..19..22...25...28...31...34...37...40...43...46...49...52
.1..6..12..18..24..30..36..42...48...54...60...66...72...78...84...90...96..102
.1..8..19..30..41..52..63..74...85...96..107..118..129..140..151..162..173..184
.1.14..39..65..91.117.143.169..195..221..247..273..299..325..351..377..403..429
.1.20..71.128.185.242.299.356..413..470..527..584..641..698..755..812..869..926
.1.36.152.293.435.577.719.861.1003.1145.1287.1429.1571.1713.1855.1997.2139.2281
The transposed array (starting with index 0) appears as Table 2 in the Knopfmacher et al. reference. [Joerg Arndt, Aug 08 2012]


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.


EXAMPLE

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


MATHEMATICA

T[n_, k_] := 1/n*Sum[DivisorSum[n, EulerPhi[#]*(1+2*Cos[i*Pi/(k+1)])^(n/#)&], {i, 1, k}] // FullSimplify; Table[T[nk+1, k], {n, 1, 12}, {k, n, 1, 1}] // Flatten (* JeanFrançois Alcover, Dec 05 2015, adapted from PARI *)


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)));
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 */


CROSSREFS

Column 2 is A000031, col. 3 is A208772, col. 4 is A208773, col. 5 is A208774, col. 6 is A208775, col. 7 is A208776.
Sequence in context: A278492 A128139 A208721 * A104732 A132108 A210489
Adjacent sequences: A208774 A208775 A208776 * A208778 A208779 A208780


KEYWORD

nonn,tabl


AUTHOR

R. H. Hardin, Mar 01 2012


STATUS

approved



