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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. 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 (list; table; graph; refs; listen; history; text; internal format)
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[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-Fran├žois 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

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Last modified November 23 18:56 EST 2017. Contains 295128 sequences.