0,13

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

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

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

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

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,

....

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

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

(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 */

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

nonn,tabl,nice

Christian G. Bower, Nov 15 1999

approved