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A053656
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Number of cyclic graphs with oriented edges on n nodes (up to symmetry of dihedral group).
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4
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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; internal format)
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OFFSET
| 1,2
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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.)
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REFERENCES
| 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, A stability result for a Hilbert series of O_n(C) invariants.
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LINKS
| Jeb F. Willenbring, Home page
Index entries for sequences related to bracelets
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FORMULA
| Sum_{n >= 1} \left( \frac{x^{2n}}{2(1-x^2)^n}+\frac{x^{2n}}{2n} Sum_{ d divides n } \frac{\phi(d)}{(1-x^d)^\frac{2n}{d}} \right)
a(n) = A000031(n)/2 + (if n even) 2^(n/2-2).
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EXAMPLE
| 2 at n=3 because there are two such cycles. On (o -> o -> o ->) and (o -> o <- o ->).
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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)
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MATHEMATICA
| a[n_] := Total[ EulerPhi[#]*2^(n/#)& /@ Divisors[n]]/(2n) + 2^(n/2-2)(1-Mod[n, 2]); Table[a[n], {n, 1, 35}](* From Jean-François Alcover, Nov 21 2011 *)
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CROSSREFS
| Sequence in context: A072488 A074818 A110199 * A035054 A099537 A109525
Adjacent sequences: A053653 A053654 A053655 * A053657 A053658 A053659
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Jeb F. Willenbring (jwillenb(AT)ucsd.edu), Feb 14 2000
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EXTENSIONS
| More terms and additional comments from Christian G. Bower (bowerc(AT)usa.net), Dec 13 2001
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