A214309 a(n) is the number of representative four-color bracelets (necklaces with turning over allowed) with n beads, for n >= 4.

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

Author

Wolfdieter Lang, Jul 31 2012

Keywords

• nonn

Comments

This is the fourth column (m=4) of triangle A213940.

The relevant p(n,4)= A008284(n,4) representative color multinomials have exponents (signatures) from the 4 part partitions of n, written with nonincreasing parts. E.g., n=6: [3,1,1,1] and [2,2,1,1] (p(6,4)=2). The corresponding representative bracelets have the four-color multinomials c[1]^3*c[2]*c[3]*c[4] and c[1]^2*c[2]^2*c[3]*c[4].

Compare this with A032275 where also bracelets with less than four colors are included, and not only representatives are counted.

Number of n-length bracelets w over a 4-ary alphabet {a1,a2,...,a4} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a4) >= 1, where #(w,x) counts the letters x in word w (bracelet analog of A226883). The number of 4 color bracelets up to permutations of colors is given by A056359. - Andrew Howroyd, Sep 26 2017

Examples

			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.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 4..200

Crossrefs

Cf. A213939, A213940, A214311 (m=5), A214312 (m=4, all bracelets).

Cf. A056359, A226883.

Programs

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], [4,4]); t/2} \\ Andrew Howroyd, Sep 26 2017

Formula

a(n) = A213940(n,4), n >= 4.

a(n) = sum(A213939(n,k),k=(2+floor(n/2) + p(n,3))..(p(n,4)+1+floor(n/2)+p(n,3))), n>=4, with p(n,m) = A008284(n,m) the number of partitions of n with m parts.

Extensions

Terms a(26) and beyond from Andrew Howroyd, Sep 26 2017