A005656 Number of bracelets (turn over necklaces) with n red, 1 pink and n - 3 blue beads; also reversible strings with n red and n-3 blue beads. (Formerly M2920)

0, 0, 1, 3, 12, 44, 170, 651, 2520, 9752, 37854, 147070, 572264, 2229096, 8692788, 33933459, 132594480, 518584880, 2029976630, 7952706234, 31179618184, 122331419080, 480283635468, 1886828198398

Offset

1,4

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Marcia Ascher, Mu torere: an analysis of a Maori game, Math. Mag. 60 (1987), no. 2, 90-100.

R. K. Guy & N. J. A. Sloane, Correspondence, 1985

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]

Index entries for sequences related to bracelets

Formula

a(n) = (1/2)*(binomial(2*n - 3, n - 3) + binomial(n - 2, floor((n - 3)/2))). - Michael Somos

Maple

A005656:=n->(1/2)*(binomial(2*n-3, n-3) + binomial(n-2, floor((n-3)/2))): seq(A005656(n), n=1..30); # Wesley Ivan Hurt, Oct 06 2017

Mathematica

Table[(1/2) (Binomial[2 n - 3, n - 3] + Binomial[n - 2, Floor[(n - 3) / 2]]), {n, 40}] (* Vincenzo Librandi, Oct 08 2017 *)

Prog

(PARI) C(n, k)= if(k<0||k>n, 0, n!/k!/(n-k)!);
a(n)= (1/2) *(C(2*n-3, n-3)+C(n-2, (n-3)\2));
(Magma) [(1/2)*(Binomial(2*n-3, n-3) + Binomial(n-2, Floor((n-3)/2))): n in [1..30]]; // Vincenzo Librandi, Oct 08 2017

Crossrefs

a(n) = A034851(2n-3, n-3).

Sequence in context: A296225 A109437 A331473 * A339066 A260146 A229936

Adjacent sequences: A005653 A005654 A005655 * A005657 A005658 A005659

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Sequence corrected, extended and description corrected by Christian G. Bower

Status

approved