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A115118
Number of imprimitive (periodic) n-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed.
3
0, 0, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 10, 1, 11, 5, 20, 1, 36, 1, 58, 11, 95, 1, 196, 4, 317, 30, 598, 1, 1153, 1, 2068, 95, 3857, 13, 7488, 1, 13799, 317, 26288, 1, 50531, 1, 95422, 1124, 182363, 1, 351764, 10, 671144, 3857, 1290874, 1, 2492820, 97, 4794104, 13799, 9256397, 1, 17923218, 1, 34636835, 49968, 67110932, 319
OFFSET
0,5
COMMENTS
a(p) = 1 for prime p. Presumably a(n) = A115121(n) = A066656(n)/2 for odd n.
LINKS
FORMULA
a(n) = A000013(n) - A000048(n).
MATHEMATICA
a[n_] := If[n == 0, 0, Sum[EulerPhi[2d] 2^(n/d) - Boole[OddQ[d]] MoebiusMu[d] 2^(n/d), {d, Divisors[n]}]/(2n)];
Array[a, 66, 0] (* Jean-François Alcover, Aug 29 2019 *)
PROG
(PARI) a(n) = if (n==0, 0, (sumdiv(n, d, eulerphi(2*d) * 2^(n/d)) - sumdiv(n, d, (d%2)*(moebius(d)*2^(n/d))))/(2*n)); \\ Michel Marcus, Oct 21 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Valery A. Liskovets, Jan 17 2006
EXTENSIONS
More terms from Antti Karttunen, Oct 21 2017
STATUS
approved