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a(n) is the number of representative three-color bracelets (necklaces with turn over allowed) with n beads for n >= 3.
5

%I #18 Sep 26 2017 14:46:22

%S 1,2,6,20,40,106,304,731,1936,5769,14343,39583,117957,305576,855474,

%T 2565922,6793516,19242857,57827068,155681341,444461623,1337436721,

%U 3645877447,10471728930,31534868169,86818242806,250543852080,754851821246,2094887707000

%N a(n) is the number of representative three-color bracelets (necklaces with turn over allowed) with n beads for n >= 3.

%C This is the third column (m=3) of triangle A213940.

%C The relevant p(n,3)= A008284(n,3) representative color multinomials have exponents (signatures) from the 3 part partitions of n, written with nonincreasing parts. E.g., n=6: [4,1,1], [3,2,1] and [2,2,2] (p(6,3)=3). The corresponding representative bracelets have the three-color multinomials c[1]^4*c[2]*c[3], c[1]^3*c[2]^2*c[3] and c[1]^2*c[2]^2*c[3]^2. Therefore, color c[1] is dominant, except for the last case.

%C Compare this with A027671 where also bracelets with less than three colors are included and not only three-color representatives are counted.

%C Number of n-length bracelets w over ternary alphabet {a,b,c} such that #(w,a) >= #(w,b) >= #(w,c) >= 1, where #(w,x) counts the letters x in word w (bracelet analog of A226882). The number of 3 color bracelets up to permutations of colors is given by A056358. - _Andrew Howroyd_, Sep 26 2017

%H Andrew Howroyd, <a href="/A214307/b214307.txt">Table of n, a(n) for n = 3..200</a>

%F a(n) = A213940(n,3), n >= 3.

%F a(n) = sum(A213939(n,k),k=(2+floor(n/2))..p(n,3)+1+floor(n/2)), n >= 3, with p(n,3) = A008284(n,3) the number of partitions of n with three parts.

%e a(5) = A213939(5,4) + A213939(5,5) = 2 + 4 = 6 from the representative bracelets (with colors j for c[j], j=1,2,3) 11123, 11213, 11223, 11232, 12123 and 12213 , all taken cyclically. The first two have color signature (exponents) [3,1,1] and the other four ones have signature [2,2,1].

%e a(6) = A213939(6,5) + A213939(6,6) + A213939(6,7) = 3 + 6 + 11 = 20. The first three representative bracelets have color signature [4,1,1], the next six have signature [3,2,1] and the remaining 11 ones have signature [2,2,2]. The corresponding representative color multinomials are c[1]^4*c[2]*c[3], c[1]^3*c[2]^2*c[3] and c[1]^2*c[2]^2*c[3]^2.

%o (PARI)

%o Cyc(v)={my(g=fold(gcd,v), s=vecsum(v)); sumdiv(g, d, eulerphi(d)*(s/d)!/prod(i=1, #v, (v[i]/d)!))/s}

%o CPal(v)={my(odds=#select(t->t%2,v), s=vecsum(v)); if(odds>2, 0, ((s-odds)/2)!/prod(i=1, #v, (v[i]\2)!))}

%o a(n)={my(t=0); forpart(p=n, t+=Cyc(Vec(p))+CPal(Vec(p)), [1,n], [3,3]); t/2} \\ _Andrew Howroyd_, Sep 26 2017

%Y Cf. A213939, A213940, A214309 (m=4), A214310 (m=3, all bracelets).

%Y Cf. A056358, A226882.

%K nonn

%O 3,2

%A _Wolfdieter Lang_, Jul 31 2012

%E Terms a(26) and beyond from _Andrew Howroyd_, Sep 26 2017