A045678 -id:A045678 - 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

A045687 Number of 2n-bead balanced binary necklaces of fundamental period 2n which are equivalent to their reversed complement, but are not equivalent to their reverse and complement.

Original entry on oeis.org

0, 0, 0, 2, 4, 12, 24, 56, 112, 238, 480, 992, 1980, 4032, 8064, 16242, 32512, 65280, 130536, 261632, 523260, 1047494, 2095104, 4192256, 8384400, 16773108, 33546240, 67100432, 134201340, 268419072, 536837640, 1073709056, 2147418112

Offset: 0

David W. Wilson

nonn

The number of length 2n balanced binary Lyndon words which are equivalent to their reversed complement is A000740(n) and the number which are equivalent to their reverse, complement and reversed complement is A045683(n). - Andrew Howroyd, Sep 28 2017

Joerg Arndt, Matters Computational (The Fxtbook), p.119
Index entries for sequences related to Lyndon words

Cf. A000740, A045669, A045678, A045683.

a740[n_] := DivisorSum[n, MoebiusMu[n/#]*2^(#-1)&];
a45674[0] = 1; a45674[n_] := Module[{t = 0, r = n}, While[EvenQ[r], r = Quotient[r, 2]; t += 2^(r-1)]; t + 2^Quotient[r, 2]];
a45683[0] = 1; a45683[n_] := DivisorSum[n, MoebiusMu[n/#]*a45674[#]&];
a[0] = 0; a[n_] := a740[n] - a45683[n];
Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Sep 30 2017, after Andrew Howroyd *)

From Andrew Howroyd, Sep 28 2017: (Start)
Moebius transform of A045678.
a(n) = A000740(n) - A045683(n).
(End)

Incorrect formulas and comments removed by Andrew Howroyd, Sep 28 2017

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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A045687 Number of 2n-bead balanced binary necklaces of fundamental period 2n which are equivalent to their reversed complement, but are not equivalent to their reverse and complement.

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