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A052307
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Triangle read by rows: T(n,k) = number of bracelets (reversible necklaces) with n beads, k of which are black and n-k are white.
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14
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 3, 4, 4, 3, 1, 1, 1, 1, 4, 5, 8, 5, 4, 1, 1, 1, 1, 4, 7, 10, 10, 7, 4, 1, 1, 1, 1, 5, 8, 16, 16, 16, 8, 5, 1, 1, 1, 1, 5, 10, 20, 26, 26, 20, 10, 5, 1, 1, 1, 1, 6, 12, 29, 38, 50, 38, 29, 12, 6, 1, 1, 1, 1, 6
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,13
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COMMENTS
| Equivalently, T(n,k) is the number of orbits of k-element subsets of the vertices of a regular n-gon under the usual action of the dihedral group D_n, or under the action of Euclidean plane isometries. Note that each row of the table is symmetric and unimodal. [From Austin Shapiro (auspex(AT)umich.edu), Apr 20 2009]
Also, the number of k-chords in n-tone equal temperament, up to (musical) transposition and inversion. Example: there are 29 tetrachords, 38 pentachords, 50 hexachords in the familiar 12-tone equal temperament. Called "Forte set-classes," after Allen Forte who first catalogued them. - Jon Wild, May 21, 2004.
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REFERENCES
| N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.
Martin Gardner, "New Mathematical Diversions from Scientific American" (Simon and Schuster, New York, 1966), pages 245-246.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
Hansraj Gupta, Indian J. pure appl. Math. 10 (8) (1979) 964-999.
A. L. Patterson, Ambiguities in the X-Ray Analysis of Crystal Structures, Phys. Rev. 65 (1944), 195 - 201 (see Table I). [From N. J. A. Sloane, Mar 14 2009]
V. S. Shevelev, Necklaces and convex k-Gons, Ind. J. pure appl. Math 35 (5) (2004) 629-638
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LINKS
| Index entries for sequences related to bracelets
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EXAMPLE
| Triangle begins:
.1,
.1,1,
.1,1,1,
.1,1,1,1,
.1,1,2,1,1,
.1,1,2,2,1,1,
.1,1,3,3,3,1,1,
.1,1,3,4,4,3,1,1,
.1,1,4,5,8,5,4,1,1,
.1,1,4,7,10,10,7,4,1,1,
.1,1,5,8,16,16,16,8,5,1,1,
.1,1,5,10,20,26,26,20,10,5,1,1,
.1,1,6,12,29,38,50,38,29,12,6,1,1,
....
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MAPLE
| A052307 := proc(n, k)
local hk, a, d;
if k = 0 then
return 1 ;
end if;
hk := k mod 2 ;
a := 0 ;
for d in numtheory[divisors](igcd(k, n)) do
a := a+ numtheory[phi](d)*binomial(n/d-1, k/d-1) ;
end do:
%/k + binomial(floor((n-hk)/2), floor(k/2)) ;
%/2 ;
end proc: # R. J. Mathar, Sep 04 2011
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MATHEMATICA
| Table[If[m*n===0, 1, 1/2*If[EvenQ[n], If[EvenQ[m], Binomial[n/2, m/2], Binomial[(n-2)/2, (m-1)/2 ]], If[EvenQ[m], Binomial[(n-1)/2, m/2], Binomial[(n-1)/2, (m-1)/2]]] + 1/2*Fold[ #1 +(EulerPhi[ #2]*Binomial[n/#2, m/#2])/n &, 0, Intersection[Divisors[n], Divisors[m]]]], {n, 0, 12}, {m, 0, n}] - Wouter Meeussen Aug 05 2002, Jan 19 2009
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CROSSREFS
| Row sums: A000029. Columns 0-12: A000012, A000012, A026809, A005232, A032279, A005513, A032280, A005514, A032281, A005515, A032282, A005516.
Cf. A047996, A051168, A052308-A052310.
Sequence in context: A140356 A119963 A057790 * A067059 A049704 A047996
Adjacent sequences: A052304 A052305 A052306 * A052308 A052309 A052310
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KEYWORD
| nonn,tabl,nice
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AUTHOR
| Christian G. Bower (bowerc(AT)usa.net), Nov 15 1999.
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