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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. 27
 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, 14, 35, 57, 76, 76, 57, 35, 14, 6, 1, 1 (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. N. Zagaglia Salvi, Ordered partitions and colourings of cycles and necklaces, Bull. Inst. Combin. Appl., 27 (1999), 37-40. 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:1405.3616 [cond-mat.stat-mech], 2014. See Section V. H. Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10(8) (1979), 964-999. S. Karim, J. Sawada, Z. Alamgirz, and S. M. Husnine, Generating bracelets with fixed content, Theoretical Computer Science, 475 (2013), 103-112. John P. McSorley and Alan H. Schoen, Rhombic tilings of (n, k)-ovals, (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] Richard H. Reis, A formula for C(T) in Gupta's paper, Indian J. Pure and Appl. Math., 10(8) (1979), 1000-1001. V. Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35(5) (2004), 629-638. V. Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35(5) (2004), 629-638. Index entries for sequences related to bracelets FORMULA T(0,0) = 1. If n > 0, T(n,k) = binomial(floor(n/2) - (k mod 2) * (1 - (n mod 2)), floor(k/2)) / 2 + Sum_{d|n, d|k} (phi(d)*binomial(n/d, k/d)) / (2*n). - Washington Bomfim, Jun 30 2012 [edited by Petros Hadjicostas, May 29 2019] From Freddy Barrera, Apr 21 2019: (Start) T(n,k) = (1/2) * (A119963(n,k) + A047996(n,k)). T(n,k) = T(n, n-k) for each k < n (Theorem 2 of H. Gupta). (End) G.f. for column k >= 1: (x^k/2) * ((1/k) * Sum_{m|k} phi(m)/(1 - x^m)^(k/m) + (1 + x)/(1 - x^2)^floor((k/2) + 1)). (This formula is due to Herbert Kociemba.) - Petros Hadjicostas, May 25 2019 Bivariate o.g.f.: Sum_{n,k >= 0} T(n, k)*x^n*y^k = (1/2) * ((x + 1) * (x*y + 1) / (1 - x^2 * (y^2 + 1)) + 1 - Sum_{d >= 1} (phi(d)/d) * log(1 - x^d * (1 + y^d))). - Petros Hadjicostas, Jun 13 2019 EXAMPLE Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) 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 */ (Python) from sympy import binomial as C, totient, divisors, gcd def T(n, k): return 1 if n==0 else C((n//2) - k%2 * (1 - n%2), (k//2))/2 + sum(totient(d)*C(n//d, k//d) for d in divisors(gcd(n, k)))/(2*n) for n in range(11): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Apr 23 2017 CROSSREFS Row sums: A000029. Columns 0-12: A000012, A000012, A008619, A001399, A005232, A032279, A005513, A032280, A005514, A032281, A005515, A032282, A005516. Cf. A047996, A051168, A052308, A052309, A052310. Sequence in context: A057790 A350889 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 September 29 02:45 EDT 2023. Contains 365749 sequences. (Running on oeis4.)