A213942 a(n) is the number of representative two-color bracelets (necklaces with turnover allowed) with n beads for n >= 2.

1, 1, 3, 3, 7, 8, 18, 22, 46, 62, 136, 189, 409, 611, 1344, 2055, 4535, 7154, 15881, 25481, 56533, 92204, 204759, 337593, 748665, 1246862, 2762111, 4636389, 10253938, 17334800, 38278784, 65108061, 143534770, 245492243, 540353057, 928772649, 2041154125

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

a(n) = A213940(n,2), n >= 2.

a(n) = Sum_{k=2..A008284(n,2)+1} A213939(n,k), n >= 2, with A008284(n,2) = floor(n/2).

a(2n) = (A000029(2n) + A005648(n)) / 2 - 1, a(2n+1) = A000029(2n+1) / 2 - 1. - Andrew Howroyd, Sep 25 2017

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

Cf. A213939, A213940, A214307 (m=3), A214308 (m=2, all bracelets).

Cf. A056357, A226881.

Sequence in context: A116157 A056357 A288728 * A144554 A177936 A337975

Adjacent sequences: A213939 A213940 A213941 * A213943 A213944 A213945

Keyword

nonn

Author

Wolfdieter Lang, Jul 31 2012

Extensions

Terms a(26) and beyond from Andrew Howroyd, Sep 25 2017

Status

approved