cp's OEIS Frontend

A364468 Number of primitive n-bead necklaces (turning over is allowed) comprising elements of two flavors where complements and scalings are equivalent.

%I A364468 #18 Jul 27 2023 12:25:33
%S A364468 1,1,1,1,1,3,3,8,12,20,35,62,106,189,343,603,1130,2055,3860,7154,
%T A364468 13562,25463,48607,92204,176646,337587,649151,1246819,2404519,4636389,
%U A364468 8963497,17334800,33585928,65107935,126385919,245492221,477345359,928772649,1808662015,3524337599,6872457828,13409202675,26179870365
%N A364468 Number of primitive n-bead necklaces (turning over is allowed) comprising elements of two flavors where complements and scalings are equivalent.
%C A364468 A complement necklace is one where the flavor of each element is inverted ("010" is equivalent to "101"). A scaled necklace is one where each element in the sequence is repeated by the same integer scalar ("010" is equivalent to "001100", "000111000", etc.).
%F A364468 a(n) = A000046(n) - Sum_{k = nontrivial divisors of n} a(k). (nontrivial divisors, d: 1 < d < n.)
%e A364468 For a(4) = 1, there is one solution: "1110". The other primitive sequence "1100" can be reduced to "10", which no longer uses 4 elements.
%e A364468 For a(6) = 3, there are three solutions: "111110", "111010", and "110010". The other primitive sequences "111100" and "111000" can be reduced to "110" and "10", respectively, which no longer use 6 elements.
%o A364468 (PARI)
%o A364468 a11(n) = if( n<1, n==0, 2^(n\2) / 2 + sumdiv(n, k, eulerphi(2*k) * 2^(n/k)) / (4*n));
%o A364468 a46(n) = {
%o A364468   my(s=0);
%o A364468   fordiv (n, d,
%o A364468     s+=moebius(d)*a11(n/d));
%o A364468   s};
%o A364468 a364468(n) = {
%o A364468   my(s=a46(n));
%o A364468   fordiv (n, k,
%o A364468     s-=if(k!=1&&k!=n, a364468(k), 0));
%o A364468   s};
%o A364468 for (k=1,42, print1(a364468(k),", "))  \\ _Hugo Pfoertner_, Jul 26 2023
%Y A364468 Cf. A000046, A070842, A000011.
%K A364468 nonn
%O A364468 0,6
%A A364468 _Richard B. Canty_, Jul 25 2023