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%I M0312 N0114 #100 Mar 18 2022 00:09:52
%S 1,1,2,2,4,4,8,9,18,23,44,63,122,190,362,612,1162,2056,3914,7155,
%T 13648,25482,48734,92205,176906,337594,649532,1246863,2405236,4636390,
%U 8964800,17334801,33588234,65108062,126390032,245492244,477353376,928772650,1808676326,3524337980
%N Number of n-bead necklaces (turning over is allowed) where complements are equivalent.
%C a(n) is also the number of minimal fibrations of a bidirectional n-cycle over the 2-bouquet up to precompositions with automorphisms of the n-cycle and postcomposition with automorphisms of the 2-bouquet. (Boldi et al.) - _Sebastiano Vigna_, Jan 08 2018
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A000011/b000011.txt">Table of n, a(n) for n = 0..3335</a> (first 201 terms from T. D. Noe)
%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>
%H Paolo Boldi and Sebastiano Vigna, <a href="https://doi.org/10.1016/S0012-365X(00)00455-6">Fibrations of Graphs</a>, Discrete Math., 243 (2002), 21-66.
%H H. Bottomley, <a href="/A000011/a000011_a000013.gif">Initial terms of A000011 and A000013</a>
%H Aharon Davidson, <a href="https://arxiv.org/abs/1907.03090">From Planck Area to Graph Theory: Topologically Distinct Black Hole Microstates</a>, arXiv:1907.03090 [gr-qc], 2019.
%H Daniel T. Eatough and Keith A. Seffen, <a href="https://doi.org/10.1115/1.4045422">Calculating the Fold Angles of Any Vertex Roof Using a Spherical Image Technique</a>, J. Mechanisms Robotics (2020) Vol. 12, No. 3, 031004.
%H N. J. Fine, <a href="http://projecteuclid.org/euclid.ijm/1255381350">Classes of periodic sequences</a>, Illinois J. Math., 2 (1958), 285-302.
%H Shinsaku Fujita, <a href="https://doi.org/10.1246/bcsj.20160369">alpha-beta Itemized Enumeration of Inositol Derivatives and m-Gonal Homologs by Extending Fujita's Proligand Method</a>, Bull. Chem. Soc. Jpn. 2017, 90, 343-366; doi:10.1246/bcsj.20160369. See Table 8.
%H E. N. Gilbert and J. Riordan, <a href="http://projecteuclid.org/euclid.ijm/1255631587">Symmetry types of periodic sequences</a>, Illinois J. Math., 5 (1961), 657-665.
%H W. D. Hoskins and Anne Penfold Street, <a href="http://dx.doi.org/10.1017/S1446788700017547">Twills on a given number of harnesses</a>, J. Austral. Math. Soc. Ser. A 33 (1982), no. 1, 1-15.
%H W. D. Hoskins and A. P. Street, <a href="/A005513/a005513_1.pdf">Twills on a given number of harnesses</a>, J. Austral. Math. Soc. (Series A), 33 (1982), 1-15. (Annotated scanned copy)
%H Yi Hu, <a href="https://hdl.handle.net/10161/23828">Numerical Transfer Matrix Method of Next-nearest-neighbor Ising Models</a>, Master's Thesis, Duke Univ. (2021).
%H Yi Hu and Patrick Charbonneau, <a href="https://arxiv.org/abs/2106.08442">Numerical transfer matrix study of frustrated next-nearest-neighbor Ising models on square lattices</a>, arXiv:2106.08442 [cond-mat.stat-mech], 2021.
%H Karyn McLellan, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i4p32">Periodic coefficients and random Fibonacci sequences</a>, Electronic Journal of Combinatorics, 20(4), 2013, #P32.
%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. P. Street, <a href="/A005513/a005513.pdf">Letter to N. J. A. Sloane, N.D.</a>
%H Zhe Sun, T. Suenaga, P. Sarkar, S. Sato, M. Kotani, and H. Isobe, <a href="https://doi.org/10.1073/pnas.1606530113">Stereoisomerism, crystal structures, and dynamics of belt-shaped cyclonaphthylenes</a>, Proc. Nat. Acad. Sci. USA, vol. 113 no. 29, pp. 8109-8114, doi: 10.1073/pnas.1606530113.
%H A. Yajima, <a href="http://dx.doi.org/10.1246/bcsj.20140204">How to calculate the number of stereoisomers of inositol-homologs</a>, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264; doi:10.1246/bcsj.20140204. See Tables 1 and 2 (and text).
%H <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>
%H <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>
%F a(n) = (A000013(n) + 2^floor(n/2))/2.
%e From Jason Orendorff (jason.orendorff(AT)gmail.com), Jan 09 2009: (Start)
%e The binary bracelets for small n are:
%e n: bracelets
%e 0: (the empty bracelet)
%e 1: 0
%e 2: 00, 01
%e 3: 000, 001
%e 4: 0000, 0001, 0011, 0101
%e 5: 00000, 00001, 00011, 00101
%e 6: 000000, 000001, 000011, 000101, 000111, 001001, 001011, 010101
%e (End)
%p with(numtheory): A000011 := proc(n) local s,d; if n = 0 then RETURN(1) else s := 2^(floor(n/2)); for d in divisors(n) do s := s+(phi(2*d)*2^(n/d))/(2*n); od; RETURN(s/2); fi; end;
%t a[n_] := Fold[ #1 + EulerPhi[2#2]2^(n/#2)/(2n) &, 2^Floor[n/2], Divisors[n]]/2
%t a[ n_] := If[ n < 1, Boole[n == 0], 2^Quotient[n, 2] / 2 + DivisorSum[ n, EulerPhi[2 #] 2^(n/#) &] / (4 n)]; (* _Michael Somos_, Dec 19 2014 *)
%o (PARI) {a(n) = if( n<1, n==0, 2^(n\2) / 2 + sumdiv(n, k, eulerphi(2*k) * 2^(n/k)) / (4*n))}; /* _Michael Somos_, Jun 03 2002 */
%Y Column 2 of A320748.
%Y Cf. A000013. Bisections give A000117 and A092668.
%Y The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.
%K nonn,nice,easy
%O 0,3
%A _N. J. A. Sloane_
%E Better description from _Christian G. Bower_
%E More terms from _David W. Wilson_, Jan 13 2000
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