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%I #54 Apr 29 2024 15:32:59
%S 1,2,2,4,4,9,10,22,30,62,94,192,316,623,1096,2122,3856,7429,13798,
%T 26500,49940,95885,182362,350650,671092,1292762,2485534,4797886,
%U 9256396,17904476,34636834,67126282,130150588,252679832,490853416
%N Number of cyclic graphs with oriented edges on n nodes (up to symmetry of dihedral group).
%C Also number of bracelets (or necklaces) with n red or blue beads such that the beads switch colors when bracelet is turned over.
%C 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.)
%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. (Boldi et al.) - _Sebastiano Vigna_, Jan 08 2018
%D Jeb F. Willenbring, A stability result for a Hilbert series of O_n(C) invariants.
%H Seiichi Manyama, <a href="/A053656/b053656.txt">Table of n, a(n) for n = 1..3334</a>
%H Rémi Cocou Avohou, Joseph Ben Geloun, and Reiko Toriumi, <a href="https://arxiv.org/abs/2404.16404">Counting U(N)^{⊗r} ⊗ O(N)^{⊗q} invariants and tensor model observables</a>, arXiv:2404.16404 [hep-th], 2024. See p. 18.
%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 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. See Table 8.
%H T. Pisanski, D. Schattschneider and B. Servatius, <a href="http://www.jstor.org/stable/27642932">Applying Burnside's lemma to a one-dimensional Escher problem</a>, Math. Mag., 79 (2006), 167-180.
%H Jeb F. Willenbring, <a href="https://uwm.edu/math/people/willenbring-jeb/">Home page</a>
%H A. Yajima, <a href="https://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. See Tables 1 and 2 (and text).
%H <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>
%F 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]
%F a(n) = A000031(n)/2 + (if n even) 2^(n/2-2).
%e 2 at n=3 because there are two such cycles. On (o -> o -> o ->) and (o -> o <- o ->).
%p v:=proc(n) local k, t1; t1:=0; for k in divisors(n) do t1 := t1+phi(k)*2^(n/k); od: t1; end;
%p h:=n-> if n mod 2 = 0 then (n/2)*2^(n/2); else 0; fi;
%p A053656:=n->(v(n)+h(n))/(2*n); # _N. J. A. Sloane_, Nov 11 2006
%t 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 *)
%o (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
%Y The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.
%K nonn,easy,nice,changed
%O 1,2
%A Jeb F. Willenbring (jwillenb(AT)ucsd.edu), Feb 14 2000
%E More terms and additional comments from _Christian G. Bower_, Dec 13 2001
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