1,2

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.)

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.

Jeb F. Willenbring, Home page

Index entries for sequences related to bracelets

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).

2 at n=3 because there are two such cycles. On (o -> o -> o ->) and (o -> o <- o ->).

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)

Sequence in context: A072488 A074818 A110199 this_sequence A035054 A099537 A109525

Adjacent sequences: A053653 A053654 A053655 this_sequence A053657 A053658 A053659

nonn,easy,nice

Jeb F. Willenbring (jwillenb(AT)ucsd.edu), Feb 14 2000

More terms and additional comments from Christian G. Bower (bowerc(AT)usa.net), Dec 13 2001