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A214307
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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.
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5
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1, 2, 6, 20, 40, 106, 304, 731, 1936, 5769, 14343, 39583, 117957, 305576, 855474, 2565922, 6793516, 19242857, 57827068, 155681341, 444461623, 1337436721, 3645877447, 10471728930, 31534868169, 86818242806, 250543852080, 754851821246, 2094887707000
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OFFSET
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3,2
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COMMENTS
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This is the third column (m=3) of triangle A213940.
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.
Compare this with A027671 where also bracelets with less than three colors are included and not only three-color representatives are counted.
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
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LINKS
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FORMULA
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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.
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EXAMPLE
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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].
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.
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PROG
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(PARI)
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}
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)!))}
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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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