

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 180degree 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 ncycle over the 2bouquet up to precompositions with automorphisms of the ncycle. (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), 2166.
Shinsaku Fujita, alphabeta Itemized Enumeration of Inositol Derivatives and mGonal Homologs by Extending Fujita's Proligand Method, Bull. Chem. Soc. Jpn. 2017, 90, 343366. See Table 8.
T. Pisanski, D. Schattschneider and B. Servatius, Applying Burnside's lemma to a onedimensional Escher problem, Math. Mag., 79 (2006), 167180.
Jeb F. Willenbring, Home page
A. Yajima, How to calculate the number of stereoisomers of inositolhomologs, Bull. Chem. Soc. Jpn. 2014, 87, 12601264. See Tables 1 and 2 (and text).
Index entries for sequences related to bracelets


FORMULA

G.f.: x/(1x) + x^2/(2*(12*x^2)) + Sum_{n >= 1} (x^(2*n)/(2*n)) * Sum_{ d divides n } phi(d)/(1x^d)^(2*n/d), or x^2/(2*(12*x^2))  Sum_{n >= 1} phi(n)*log(12*x^n)/(2*n). [corrected and extended by Andrey Zabolotskiy, Oct 17 2017]
a(n) = A000031(n)/2 + (if n even) 2^(n/22).


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/22)(1Mod[n, 2]); Table[a[n], {n, 1, 35}] (* JeanFrançois Alcover, Nov 21 2011 *)


PROG

(PARI) a(n)={(sumdiv(n, d, eulerphi(d)*2^(n/d))/n + if(n%2==0, 2^(n/21)))/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



