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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)
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

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

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

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.

N. J. A. Sloane, Classic Sequences

Index entries for sequences related to bracelets

FORMULA

a(n) = (1/2) (C(2n-1, n)+C(n-1, [ n/2 ])).

a(n) = A034851(2n-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

MATHEMATICA

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

PROG

(PARI) a(n)= (1/2) *(C(2*n-1, n)+C(n-1, n\2)) where C(n, k)=if(k<0|k>n, 0, n!/k!/(n-k)!)

(MAGMA) [((Binomial(2*n-1, n)+Binomial(n-1, Floor(n/2)))/2): n in [1..30]]; // Vincenzo Librandi, May 24 2012

CROSSREFS

Sequence in context: A275755 A150085 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 Formula from Michael Somos.

STATUS

approved

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Last modified December 6 06:58 EST 2016. Contains 278775 sequences.