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A208544 T(n,k) = Number of n-bead necklaces of k colors allowing reversal, with no adjacent beads having the same color. 8
1, 2, 0, 3, 1, 0, 4, 3, 0, 0, 5, 6, 1, 1, 0, 6, 10, 4, 6, 0, 0, 7, 15, 10, 21, 3, 1, 0, 8, 21, 20, 55, 24, 13, 0, 0, 9, 28, 35, 120, 102, 92, 9, 1, 0, 10, 36, 56, 231, 312, 430, 156, 30, 0, 0, 11, 45, 84, 406, 777, 1505, 1170, 498, 29, 1, 0, 12, 55, 120, 666, 1680, 4291, 5580, 4435 (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

.0.1..3...6...10....15.....21.....28......36......45......55.......66.......78

.0.0..1...4...10....20.....35.....56......84.....120.....165......220......286

.0.1..6..21...55...120....231....406.....666....1035....1540.....2211.....3081

.0.0..3..24..102...312....777...1680....3276....5904....9999....16104....24882

.0.1.13..92..430..1505...4291..10528...23052...46185...86185...151756...254618

.0.0..9.156.1170..5580..19995..58824..149796..341640..714285..1391940..2559414

.0.1.30.498.4435.25395.107331.365260.1058058.2707245.6278140.13442286.26942565

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 264 terms from R. H. Hardin)

FORMULA

T(2n+1,k) = A208535(2n+1,k)/2 for n > 0, T(2n,k) = (A208535(2n,k) + (k*(k-1)^n)/2)/2. - Andrew Howroyd, Mar 12 2017

Empirical for row n:

n=1: a(k) = k

n=2: a(k) = (1/2)*k^2 - (1/2)*k

n=3: a(k) = (1/6)*k^3 - (1/2)*k^2 + (1/3)*k

n=4: a(k) = (1/8)*k^4 - (1/4)*k^3 + (3/8)*k^2 - (1/4)*k

n=5: a(k) = (1/10)*k^5 - (1/2)*k^4 + k^3 - k^2 + (2/5)*k

n=6: a(k) = (1/12)*k^6 - (1/2)*k^5 + (3/2)*k^4 - (7/3)*k^3 + (23/12)*k^2 - (2/3)*k

n=7: a(k) = (1/14)*k^7 - (1/2)*k^6 + (3/2)*k^5 - (5/2)*k^4 + (5/2)*k^3 - (3/2)*k^2 + (3/7)*k

EXAMPLE

All solutions for n=7, k=3:

..1....1....1....1....1....1....1....1....1

..2....2....2....2....2....2....2....2....2

..3....3....1....1....3....1....3....1....3

..1....1....2....2....1....2....2....3....2

..2....3....3....3....3....1....3....1....3

..3....1....1....2....2....2....2....2....1

..2....3....3....3....3....3....3....3....3

MATHEMATICA

T[n_, k_] := If[n == 1, k, (DivisorSum[n, EulerPhi[n/#]*(k-1)^#&]/n + If[ OddQ[n], 1-k, k*(k-1)^(n/2)/2])/2]; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-Fran├žois Alcover, Oct 30 2017, after Andrew Howroyd *)

PROG

(PARI)

T(n, k) = if(n==1, k, (sumdiv(n, d, eulerphi(n/d)*(k-1)^d)/n + if(n%2, 1-k, k*(k-1)^(n/2)/2))/2);

for(n=1, 10, for(k=1, 10, print1(T(n, k), ", ")); print) \\ Andrew Howroyd, Oct 14 2017

CROSSREFS

Cf. A081720, A208535, A106512.

Main diagonal is A208538.

Columns 3..7 are A208539, A208540, A208541, A208542, A208543.

Row 2 is A000217(n-1).

Row 3 is A000292(n-2).

Row 4 is A002817(n-1).

Row 5 is A164938(n-1).

Row 6 is A027670(n-1).

Sequence in context: A185914 A144257 A257232 * A208535 A284856 A276550

Adjacent sequences:  A208541 A208542 A208543 * A208545 A208546 A208547

KEYWORD

nonn,tabl

AUTHOR

R. H. Hardin, Feb 27 2012

STATUS

approved

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Last modified February 20 21:53 EST 2018. Contains 299387 sequences. (Running on oeis4.)