A005654 Number of bracelets (turn over necklaces) with n red, 1 pink and n-1 blue beads; also reversible strings with n red and n-1 blue beads; also next-to-central column in Losanitsch's triangle A034851. (Formerly M1640)

1, 2, 6, 19, 66, 236, 868, 3235, 12190, 46252, 176484, 676270, 2600612, 10030008, 38781096, 150273315, 583407990, 2268795980, 8836340260, 34461678394, 134564560988, 526024917288, 2058358034616, 8061901596814, 31602652961516, 123979635837176, 486734861612328

Offset

1,2

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

A. Ivanyi, L. Lucz, T. Matuszka, and S. Pirzada, Parallel enumeration of degree sequences of simple graphs, Acta Univ. Sapientiae, Informatica, 4, 2 (2012) 260-288.

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]

N. J. A. Sloane, Classic Sequences

Index entries for sequences related to bracelets

Formula

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

a(n) = A034851(2*n-1, n-1).

Conjecture: n*(n-2)*a(n) - (5*n-3)*(n-2)*a(n-1) + 4*(n-2)*a(n-2) + 4*(5*n^2-27*n+37)*a(n-3) - 8*(2*n-7)*(n-4)*a(n-4) = 0. - R. J. Mathar, Nov 09 2013

Maple

A005654:=n->(1/2)*(binomial(2*n-1, n)+binomial(n-1, floor(n/2))): seq(A005654(n), n=1..40); # Wesley Ivan Hurt, Jan 29 2017

Mathematica

Table[(Binomial[2n-1, n]+Binomial[n-1, Floor[n/2]])/2, {n, 30}] (* Harvey P. Dale, May 17 2012 *)

Prog

(PARI) C(n, k)=binomial(n, k)
a(n)=(1/2)*(C(2*n-1, n)+C(n-1, n\2))
(Magma) [((Binomial(2*n-1, n)+Binomial(n-1, Floor(n/2)))/2): n in [1..30]]; // Vincenzo Librandi, May 24 2012

Crossrefs

Cf. A034851.

Sequence in context: A150085 A306175 A266466 * A150086 A150087 A150088

Adjacent sequences: A005651 A005652 A005653 * A005655 A005656 A005657

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Sequence extended and description corrected by Christian G. Bower

Status

approved