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.

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

Author

N. J. A. Sloane

Keywords

• nonn

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

Crossrefs

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

Programs

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

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 *)

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));

Formula

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

Extensions

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