A045675 -id:A045675 - cp's OEIS frontend

A045676 Number of 2n-bead balanced binary necklaces which are equivalent to their reverse, but not equivalent to their complement and reversed complement.

0, 0, 0, 0, 2, 2, 14, 12, 58, 54, 232, 220, 886, 860, 3360, 3304, 12730, 12614, 48348, 48108, 184224, 183732, 704376, 703384, 2702070, 2700060, 10396440, 10392408, 40108336, 40100216, 155101008, 155084752, 601047482, 601014854, 2333540428

Offset: 0

David W. Wilson

nonn

The number of 2n-bead balanced binary necklaces equivalent to their reverse is A128014(n) and those equivalent to their reverse, complement and reversed complement is A045674(n). - Andrew Howroyd, Sep 28 2017

Index entries for sequences related to necklaces

Cf. A045674, A045675, A045677, A045678, A128014.

A045674[n_] := A045674[n] = If[n == 0, 1, If[EvenQ[n], 2^(n/2 - 1) + A045674[n/2], 2^((n-1)/2)]];
a[n_] := SeriesCoefficient[(1+x)/Sqrt[1 - 4 x^2], {x, 0, n}] - A045674[n];
a /@ Range[0, 34] (* Jean-François Alcover, Sep 13 2019 *)

a(n) = A128014(n) - A045674(n). - Andrew Howroyd, Sep 28 2017

A045677 Number of 2n-bead balanced binary necklaces which are equivalent to their complement, but not equivalent to their reverse and their reversed complement.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 2, 8, 14, 36, 62, 142, 252, 524, 968, 1928, 3600, 7044, 13286, 25740, 48916, 94364, 180314, 347630, 666996, 1286712, 2477342, 4785824, 9240012, 17880320, 34604066, 67078024, 130085052, 252583200, 490722344, 954313264

Offset: 0

David W. Wilson

nonn

The number of 2n-bead balanced binary necklaces which are equivalent to their complement is A000013(n) and those which are equivalent to their reverse, complement and reversed complement is A045674(n). - Andrew Howroyd, Sep 28 2017

Index entries for sequences related to necklaces

Cf. A000013, A045674, A045675, A045676, A045678.

A045674[n_] := A045674[n] = If[n == 0, 1, If[EvenQ[n], 2^(n/2 - 1) + A045674[n/2], 2^((n - 1)/2)]];
a[n_] := If[n == 0, 1, Sum[EulerPhi[2 d] 2^(n/d), {d, Divisors[n]}]/(2 n)] - A045674[n];
a /@ Range[0, 36] (* Jean-François Alcover, Sep 13 2019 *)

a(n) = A000013(n) - A045674(n). - Andrew Howroyd, Sep 28 2017

A045684 Number of 2n-bead balanced binary necklaces of fundamental period 2n which are inequivalent to their reverse, complement and reversed complement.

Original entry on oeis.org

0, 0, 0, 0, 0, 8, 32, 168, 616, 2380, 8464, 30760, 109612, 394816, 1420616, 5149940, 18736128, 68553728, 251899620, 929814984, 3445425136, 12814382452, 47817520376, 178982546512, 671813585080, 2528191984496, 9536849432000

Offset: 0

David W. Wilson

nonn

The number of length 2n balanced binary Lyndon words is A022553(n) and the number which are equivalent to their reverse, complement and reversed complement are respectively A045680(n), A000048(n) and A000740(n). - Andrew Howroyd, Sep 29 2017

Jean-François Alcover, Table of n, a(n) for n = 0..100
Index entries for sequences related to Lyndon words

Cf. A000048, A000740, A022553, A045675, A045680, A045683.

a22553[n_] := If[n == 0, 1, Sum[MoebiusMu[n/d]*Binomial[2d, d], {d, Divisors[n]}]/(2n)];
a45680[n_] := If[n == 0, 1, DivisorSum[n, MoebiusMu[n/#] Binomial[# - Mod[#, 2], Quotient[#, 2]] &]];
a48[n_] := If[n == 0, 1, Total[MoebiusMu[#]*2^(n/#)& /@ Select[Divisors[n], OddQ]]/(2n)];
a740[n_] := Sum[MoebiusMu[n/d]*2^(d - 1), {d, Divisors[n]}];
b[n_] := Module[{t = 0, r = n}, If[n == 0, 1, While[EvenQ[r], r = Quotient[r, 2]; t += 2^(r - 1)]]; t + 2^Quotient[r, 2]];
a45683[n_] := If[n == 0, 1, DivisorSum[n, MoebiusMu[n/#]*b[#] &]];
a[n_] := If[n == 0, 0, a22553[n] - a45680[n] - a48[n] - a740[n] + 2 a45683[n]];
a /@ Range[0, 100] (* Jean-François Alcover, Sep 23 2019 *)

From Andrew Howroyd, Sep 28 2017: (Start)
Moebius transform of A045675.
a(n) = A022553(n) - A045680(n) - A000048(n) - A000740(n) + 2*A045683(n).
(End)

cp's OEIS Frontend

A045676 Number of 2n-bead balanced binary necklaces which are equivalent to their reverse, but not equivalent to their complement and reversed complement.

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A045677 Number of 2n-bead balanced binary necklaces which are equivalent to their complement, but not equivalent to their reverse and their reversed complement.

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Formula

A045684 Number of 2n-bead balanced binary necklaces of fundamental period 2n which are inequivalent to their reverse, complement and reversed complement.

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Mathematica

Formula