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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)
2

%I M2920

%S 0,0,1,3,12,44,170,651,2520,9752,37854,147070,572264,2229096,8692788,

%T 33933459,132594480,518584880,2029976630,7952706234,31179618184,

%U 122331419080,480283635468,1886828198398

%N 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.

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

%H Vincenzo Librandi, <a href="/A005656/b005656.txt">Table of n, a(n) for n = 1..1000</a>

%H Marcia Ascher, <a href="http://www.jstor.org/stable/2690304">Mu torere: an analysis of a Maori game</a>, Math. Mag. 60 (1987), no. 2, 90-100.

%H R. K. Guy & N. J. A. Sloane, <a href="/A005648/a005648.pdf">Correspondence, 1985</a>

%H F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>

%H F. Ruskey, <a href="/A000011/a000011.pdf">Necklaces, Lyndon words, De Bruijn sequences, etc.</a> [Cached copy, with permission, pdf format only]

%H <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>

%F a(n) = (1/2)*(binomial(2*n - 3, n - 3) + binomial(n - 2, floor((n - 3)/2))). - _Michael Somos_

%p 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

%t Table[(1/2) (Binomial[2 n - 3, n - 3] + Binomial[n - 2, Floor[(n - 3) / 2]]), {n, 40}] (* _Vincenzo Librandi_, Oct 08 2017 *)

%o (PARI) C(n,k)= if(k<0||k>n,0,n!/k!/(n-k)!);

%o a(n)= (1/2) *(C(2*n-3,n-3)+C(n-2,(n-3)\2));

%o (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

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

%K nonn

%O 1,4

%A _N. J. A. Sloane_

%E Sequence corrected, extended and description corrected by _Christian G. Bower_

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Last modified June 19 03:38 EDT 2021. Contains 345125 sequences. (Running on oeis4.)