

A213942


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


3



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
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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]^(n1)*c[2], c[1]^(n2)*c[2]^2, ..., c[1]^(nfloor(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 onecolor 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 twocolor 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 twocolor 5bracelets 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 anticyclic 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] (* JeanFrançois 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 A143088
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



