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 A053656 Number of cyclic graphs with oriented edges on n nodes (up to symmetry of dihedral group). 14
 1, 2, 2, 4, 4, 9, 10, 22, 30, 62, 94, 192, 316, 623, 1096, 2122, 3856, 7429, 13798, 26500, 49940, 95885, 182362, 350650, 671092, 1292762, 2485534, 4797886, 9256396, 17904476, 34636834, 67126282, 130150588, 252679832, 490853416 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Also number of bracelets (or necklaces) with n red or blue beads such that the beads switch colors when bracelet is turned over. a(n) is also the number of frieze patterns generated by filling a 1 X n block with n copies of an asymmetric motif (where the copies are chosen from original motif or a 180-degree rotated copy) and then repeating the block by translation to produce an infinite frieze pattern. (Pisanski et al.) 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. (Boldi et al.) - Sebastiano Vigna, Jan 08 2018 REFERENCES Jeb F. Willenbring, A stability result for a Hilbert series of O_n(C) invariants. LINKS Seiichi Manyama, Table of n, a(n) for n = 1..3334 Paolo Boldi and Sebastiano Vigna, Fibrations of Graphs, Discrete Math., 243 (2002), 21-66. 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. See Table 8. T. Pisanski, D. Schattschneider and B. Servatius, Applying Burnside's lemma to a one-dimensional Escher problem, Math. Mag., 79 (2006), 167-180. Jeb F. Willenbring, Home page A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264. See Tables 1 and 2 (and text). FORMULA G.f.: x/(1-x) + x^2/(2*(1-2*x^2)) + Sum_{n >= 1} (x^(2*n)/(2*n)) * Sum_{ d divides n } phi(d)/(1-x^d)^(2*n/d), or x^2/(2*(1-2*x^2)) - Sum_{n >= 1} phi(n)*log(1-2*x^n)/(2*n). [corrected and extended by Andrey Zabolotskiy, Oct 17 2017] a(n) = A000031(n)/2 + (if n even) 2^(n/2-2). EXAMPLE 2 at n=3 because there are two such cycles. On (o -> o -> o ->) and (o -> o <- o ->). MAPLE v:=proc(n) local k, t1; t1:=0; for k in divisors(n) do t1 := t1+phi(k)*2^(n/k); od: t1; end; h:=n-> if n mod 2 = 0 then (n/2)*2^(n/2); else 0; fi; A053656:=n->(v(n)+h(n))/(2*n); # N. J. A. Sloane, Nov 11 2006 MATHEMATICA a[n_] := Total[ EulerPhi[#]*2^(n/#)& /@ Divisors[n]]/(2n) + 2^(n/2-2)(1-Mod[n, 2]); Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Nov 21 2011 *) PROG (PARI) a(n)={(sumdiv(n, d, eulerphi(d)*2^(n/d))/n + if(n%2==0, 2^(n/2-1)))/2} \\ Andrew Howroyd, Jun 16 2021 CROSSREFS The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848. Sequence in context: A110199 A358429 A222736 * A035054 A099537 A109525 Adjacent sequences: A053653 A053654 A053655 * A053657 A053658 A053659 KEYWORD nonn,easy,nice AUTHOR Jeb F. Willenbring (jwillenb(AT)ucsd.edu), Feb 14 2000 EXTENSIONS More terms and additional comments from Christian G. Bower, Dec 13 2001 STATUS approved

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Last modified December 7 04:29 EST 2022. Contains 358649 sequences. (Running on oeis4.)