A214309
a(n) is the number of representative four-color bracelets (necklaces with turning over allowed) with n beads, for n >= 4.
Original entry on oeis.org
3, 6, 26, 93, 424, 1180, 4844, 16165, 66953, 216804, 852822, 2949804, 12119134, 40886724, 160826008, 572457489, 2331396595, 8104270828, 32043699894, 115995102806, 471268872328, 1674576998468, 6641876380417, 24364816845446, 98894256728960, 357006263815751
Offset: 4
a(4) = A213939(4,5) = 3 from the representative bracelets (with colors j for c[j], j=1, 2, ..., 4) 1234, 1342 and 1423, all taken cyclically. The necklace cyclic(1324), for example, becomes equivalent to cyclic(1423) under the dihedral D_4 turning over (or reflection) operation.
a(6) = A213939(6, 8) = A213939(6, 9) = 10 + 16 = 26. See the comment above for the representative color multinomials for each case.
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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], [4,4]); t/2} \\ Andrew Howroyd, Sep 26 2017
A214310
a(n) is the number of all three-color bracelets (necklaces with turning over allowed) with n beads and the three colors are from a repertoire of n distinct colors, for n >= 3.
Original entry on oeis.org
1, 24, 180, 1120, 5145, 23016, 91056, 357480, 1327095, 4893680, 17525508, 62254920, 217457695, 753332160, 2581110000, 8779264032, 29624681763, 99350001360, 331159123260, 1098168382080, 3624003213369, 11908069219816, 38972450763000, 127087400895000
Offset: 3
a(5) = A213941(5,4) + A213941(5,5) = 60 + 120 = 180 from the bracelet (with colors j for c[j], j=1, 2, ..., 5) 11123 and 11213, both taken cyclically, each representing a class of order A035206(5,4)= 30 (if all 5 colors are used), and 11223, 11232, 12123 and 12213, all taken cyclically, each representing a class of order A035206(5,5)= 30. For example, cyclic(11322) becomes equivalent to cyclic(11223) by turning over or reflection. The multiplicity A035206 depends only on the color signature.
A214313
a(n) is the number of all five-color bracelets (necklaces with turning over allowed) with n beads and the four colors are from a repertoire of n distinct colors, for n >= 5.
Original entry on oeis.org
12, 900, 25200, 442680, 5846400, 64420272, 622175400, 5466166200, 44611306740, 343916472900, 2531921456064, 17956666859040, 123458676825120, 827056125453600, 5419508203393200, 34847210197637424, 220424306985639540, 1374479672119161300, 8463477229726134000, 51536194734146965920, 310706598354410079360
Offset: 5
a(6) = A213941(6,10) = 900 from the bracelet with color signature [2,1,1,1,1] and color repertoire [c[j], j=1, 2, ..., 6]. There are A213939(6,10) = 30 bracelets with representative color multinomials c[1]^2 c[2] c[3] c[4] c[5]. If the colors c[j] are taken as j, e.g., 112345, 112354, 112435, 112453, 112534, 112543, 113245, 113254, 113425, (113452 is equivalent to 112543 by turning over), 113524, (113542 ==112453), 114235, ..., 121345, ... (all taken cyclically). Each of these 30 bracelets represents a class of A035206(6,10) = 30 bracelets when all six colors are used. Thus a(6) = 30*30 = 900 = 12*75.
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