OFFSET
2,3
COMMENTS
This is the second column (m=2) of triangle A213940.
The relevant floor(n/2) representative color multinomials are c[1]^(n-1)*c[2], c[1]^(n-2)*c[2]^2, ..., c[1]^(n-floor(n/2))* c[2]^(floor(n/2)). For such representative bracelets the color c[1] is therefore preferred. Only for even n can c[2] appear as often as c[1], namely, n/2 times.
Note that beads with different colors are always present. This is in contrast to, e.g., A000029, where not only representatives but also one-color bracelets are counted. This sequences gives the number of binary bracelets with at least as many 0's as 1's and at least one 1 (bracelet analog of A226881). The number of two-color bracelets up to permutations of colors is given by A056357. For odd n these two sequences are equal. For a(8), the bracelets 00011011 and 11100100 are equivalent in A056357 but distinct in this sequence. - Andrew Howroyd and Wolfdieter Lang, Sep 25 2017
LINKS
G. C. Greubel, Table of n, a(n) for n = 2..1000
FORMULA
EXAMPLE
a(5) = A213939(5,2) + A213939(5,3) = 1 + 2 = 3 from the representative bracelets (with colors j for c[j], j=1,2) cyclic(11112), cyclic(11122) and cyclic(11212). The first one has color signature (exponents) [4,1] and the two others have signature [3,2]. For the number of all two-color 5-bracelets with beads of five colors available see A214308(5) = 60.
a(8) = 18 = 1 + 4 + 5 + 8 for the partitions of 8 with 2 parts (7,1), (6, 2), (5,3), (4,4), respectively. see A213939(5, k), k = 2..5). The 8 representative bracelets for the exponents (signature) from partition (4,4) are B1 = (11112222), B2 = (11121222), B3 = (11212122), B4 = (11212212), B5 = (11221122), B6 = (12121212), B7 = (11122122) and B8 = (11211222). B1 to B6 are color exchange (1 <-> 2) invariant (modulo D_8 symmetry, i.e., cyclic or anti-cyclic operations). B7 is equivalent to B8 under color exchange.
This explains why A056357(8) = 17. The difference between the present sequence and A056357 is that there, besides D_n symmetry, also color exchange is allowed. Here only color exchange compatible with D_n symmetry is allowed. - Wolfdieter Lang, Sep 28 2017
MATHEMATICA
a29[n_] := (1/4)*(Mod[n, 2] + 3)*2^Quotient[n, 2] + DivisorSum[n, EulerPhi[#]*2^(n/#)&]/(2*n);
a5648[n_] := 1/2*(Binomial[2*Quotient[n, 2], Quotient[n, 2]] + DivisorSum[n, EulerPhi[#]*Binomial[2*n/#, n/#]&]/(2*n));
a[n_] := a29[n]/2 - 1 + If[EvenQ[n], a5648[n/2]/2, 0];
Array[a, 37, 2] (* Jean-François Alcover, Nov 05 2017, after Andrew Howroyd *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Wolfdieter Lang, Jul 31 2012
EXTENSIONS
Terms a(26) and beyond from Andrew Howroyd, Sep 25 2017
STATUS
approved