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A115124
Number of imprimitive (periodic) 2n-bead black-white reversible complementable necklaces with n black beads.
0
0, 0, 1, 1, 2, 1, 4, 1, 7, 3, 14, 1, 40, 1, 86, 15, 257, 1, 797, 1, 2523, 87, 8360, 1, 29218, 13, 101341, 765, 361275, 1, 1300415, 1, 4707969, 8361, 17179436, 97, 63097809, 1, 232615772, 101342, 861726044, 1, 3204597995, 1, 11955836263, 1298641, 44748176654, 1
OFFSET
0,5
COMMENTS
a(p)=1 for prime p.
FORMULA
a(n)=A006840(n) - A045633(n).
MATHEMATICA
b[n_] := (1/(2n)) DivisorSum[n, EulerPhi[n/#] Binomial[2# - 1, # - 1] + EulerPhi[2(n/#)] 2^(# - 1)&];
A006840[n_] := If[n == 0, 1, (b[n] + 2^(n - 2) + Binomial[n - Mod[n, 2], Quotient[n, 2]]/2)/2];
A045633[n_] := If[n==0, 1, Sum[MoebiusMu[n/d] A006840[d], {d, Divisors[n]}] ];
a[n_] := A006840[n] - A045633[n];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Aug 28 2019 *)
CROSSREFS
Sequence in context: A252751 A199744 A360870 * A115122 A097360 A360862
KEYWORD
easy,nonn
AUTHOR
Valery A. Liskovets, Jan 17 2006
EXTENSIONS
More terms from Jean-François Alcover, Aug 28 2019
STATUS
approved