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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

%I M1640

%S 1,2,6,19,66,236,868,3235,12190,46252,176484,676270,2600612,10030008,

%T 38781096,150273315,583407990,2268795980,8836340260,34461678394,

%U 134564560988,526024917288,2058358034616,8061901596814,31602652961516,123979635837176,486734861612328

%N 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.

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

%H Vincenzo Librandi, <a href="/A005654/b005654.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 A. Ivanyi, L. Lucz, T. Matuszka, and S. Pirzada, <a href="http://www.acta.sapientia.ro/acta-info/C4-2/info42-7.pdf">Parallel enumeration of degree sequences of simple graphs</a>, Acta Univ. Sapientiae, Informatica, 4, 2 (2012) 260-288.

%H F. Ruskey, <a href="http://web.archive.org/web/20170113110237/www.theory.cs.uvic.ca/~cos/inf/neck/NecklaceInfo.html">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 N. J. A. Sloane, <a href="/classic.html#LOSS">Classic Sequences</a>

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

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

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

%F 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

%p 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

%t Table[(Binomial[2n-1,n]+Binomial[n-1,Floor[n/2]])/2,{n,30}] (* _Harvey P. Dale_, May 17 2012 *)

%o (PARI) C(n,k)=binomial(n,k)

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

%o (MAGMA) [((Binomial(2*n-1, n)+Binomial(n-1, Floor(n/2)))/2): n in [1..30]]; // _Vincenzo Librandi_, May 24 2012

%Y Cf. A034851.

%K nonn,easy,nice

%O 1,2

%A _N. J. A. Sloane_

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

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Last modified January 22 20:48 EST 2019. Contains 319365 sequences. (Running on oeis4.)