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A000011 Number of n-bead necklaces (turning over is allowed) where complements are equivalent.
(Formerly M0312 N0114)
97
1, 1, 2, 2, 4, 4, 8, 9, 18, 23, 44, 63, 122, 190, 362, 612, 1162, 2056, 3914, 7155, 13648, 25482, 48734, 92205, 176906, 337594, 649532, 1246863, 2405236, 4636390, 8964800, 17334801, 33588234, 65108062, 126390032, 245492244, 477353376, 928772650, 1808676326, 3524337980 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..3335 (first 201 terms from T. D. Noe)
Paolo Boldi and Sebastiano Vigna, Fibrations of Graphs, Discrete Math., 243 (2002), 21-66.
Daniel T. Eatough and Keith A. Seffen, Calculating the Fold Angles of Any Vertex Roof Using a Spherical Image Technique, J. Mechanisms Robotics (2020) Vol. 12, No. 3, 031004.
N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.
Shinsaku Fujita, alpha-beta Itemized Enumeration of Inositol Derivatives and m-Gonal Homologs by Extending Fujita's Proligand Method, Bull. Chem. Soc. Jpn. 2017, 90, 343-366; doi:10.1246/bcsj.20160369. See Table 8.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
W. D. Hoskins and Anne Penfold Street, Twills on a given number of harnesses, J. Austral. Math. Soc. Ser. A 33 (1982), no. 1, 1-15.
W. D. Hoskins and A. P. Street, Twills on a given number of harnesses, J. Austral. Math. Soc. (Series A), 33 (1982), 1-15. (Annotated scanned copy)
Yi Hu and Patrick Charbonneau, Numerical transfer matrix study of frustrated next-nearest-neighbor Ising models on square lattices, arXiv:2106.08442 [cond-mat.stat-mech], 2021.
Karyn McLellan, Periodic coefficients and random Fibonacci sequences, Electronic Journal of Combinatorics, 20(4), 2013, #P32.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Zhe Sun, T. Suenaga, P. Sarkar, S. Sato, M. Kotani, and H. Isobe, Stereoisomerism, crystal structures, and dynamics of belt-shaped cyclonaphthylenes, Proc. Nat. Acad. Sci. USA, vol. 113 no. 29, pp. 8109-8114, doi: 10.1073/pnas.1606530113.
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264; doi:10.1246/bcsj.20140204. See Tables 1 and 2 (and text).
FORMULA
a(n) = (A000013(n) + 2^floor(n/2))/2.
EXAMPLE
From Jason Orendorff (jason.orendorff(AT)gmail.com), Jan 09 2009: (Start)
The binary bracelets for small n are:
n: bracelets
0: (the empty bracelet)
1: 0
2: 00, 01
3: 000, 001
4: 0000, 0001, 0011, 0101
5: 00000, 00001, 00011, 00101
6: 000000, 000001, 000011, 000101, 000111, 001001, 001011, 010101
(End)
MAPLE
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;
MATHEMATICA
a[n_] := Fold[ #1 + EulerPhi[2#2]2^(n/#2)/(2n) &, 2^Floor[n/2], Divisors[n]]/2
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 *)
PROG
(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 */
CROSSREFS
Column 2 of A320748.
Cf. A000013. Bisections give A000117 and A092668.
The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.
Sequence in context: A316624 A318770 A284613 * A187213 A022476 A347206
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
Better description from Christian G. Bower
More terms from David W. Wilson, Jan 13 2000
STATUS
approved

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Last modified March 28 11:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)