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A052307 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. 16
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; text; internal format)
OFFSET

0,13

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. [Austin Shapiro, 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

REFERENCES

Martin Gardner, "New Mathematical Diversions from Scientific American" (Simon and Schuster, New York, 1966), pages 245-246.

V. S. Shevelev, Necklaces and convex k-Gons, Ind. J. pure appl. Math 35 (5) (2004) 629-638

LINKS

Washington Bomfim, Rows n = 0..130, flattened

N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

G. Gori, S. Paganelli, A. Sharma, P. Sodano, and A. Trombettoni, Bell-Paired States Inducing Volume Law for Entanglement Entropy in Fermionic Lattices, arXiv preprint arXiv:1405.3616 #, 2014. See Section V.

H. Gupta, Enumeration of incongruent cyclic k-gons, Indian J. pure appl. Math., 10 (1979), no. 8, 964-999.

S. Karim, J. Sawada, Z. Alamgirz, and S. M. Husnine, Generating bracelets with fixed content, (2011).

John P. McSorley and Alan H. Schoen, On k-Ovals and (n, k, lambda)-Cyclic Difference Sets, and Related Topics, Discrete Math., 313 (2013), 129-154. - From N. J. A. Sloane, Nov 26 2012

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]

Index entries for sequences related to bracelets

FORMULA

T(0,0) = 1. If n > 0, T(n,k) = C([n/2] - k mod 2 * (1 - n mod 2), [k/2]) / 2 + Sum_(d|n, d|k) {Phi(d)*C(n/d, k/d)} / (2n). - Washington Bomfim, Jun 30 2012

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,

....

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

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

PROG

(PARI)

B(n, k)={ if(n==0, return(1)); GCD = gcd(n, k); S = 0;

for(d = 1, GCD, if((k%d==0)&&(n%d==0), S+=eulerphi(d)*binomial(n/d, k/d)));

return (binomial(floor(n/2)- k%2*(1-n%2), floor(k/2))/2 + S/(2*n)); }

n=0; k=0; for(L=0, 8645, print(L, " ", B(n, k)); k++; if(k>n, k=0; n++))

/* Washington Bomfim, Jun 30 2012 */

CROSSREFS

Row sums: A000029. Columns 0-12: A000012, A000012, A008619, A001399, A005232, A032279, A005513, A032280, A005514, A032281, A005515, A032282, A005516.

Cf. A047996, A051168, A052308-A052310.

Sequence in context: A119963 A057790 A224697 * A067059 A049704 A047996

Adjacent sequences:  A052304 A052305 A052306 * A052308 A052309 A052310

KEYWORD

nonn,tabl,nice

AUTHOR

Christian G. Bower, Nov 15 1999

STATUS

approved

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Last modified May 29 10:07 EDT 2015. Contains 257930 sequences.