A115122 - OEIS

A115122

Number of imprimitive (periodic) 2n-bead black-white complementable necklaces with n black beads.

0, 0, 1, 1, 2, 1, 4, 1, 7, 3, 16, 1, 49, 1, 129, 17, 415, 1, 1408, 1, 4659, 130, 16081, 1, 56858, 15, 200171, 1367, 716733, 1, 2590529, 1, 9393119, 16082, 34319668, 142, 126104312, 1, 465076161, 200172, 1723097474, 1, 6408573532, 1, 23910271229, 2587214, 89493903439

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OFFSET

0,5

COMMENTS

a(p)=1 for prime p.

LINKS

Table of n, a(n) for n=0..46.

FORMULA

a(n)=A045629(n) - A045632(n).

MATHEMATICA

A045629[n_] := If[n==0, 1, (1/(2n)) DivisorSum[n, EulerPhi[n/#] Binomial[ 2 # - 1, # - 1] + EulerPhi[2 (n/#)] 2^(# - 1)&]];

A045632[n_] := If[n==0, 1, Sum[MoebiusMu[n/d] A045629[d], {d, Divisors[n]}]];