A045665 -id:A045665 - cp's OEIS frontend

A045683 Number of 2n-bead balanced binary necklaces of fundamental period 2n which are equivalent to their reverse, complement and reversed complement.

1, 1, 1, 1, 2, 3, 3, 7, 8, 14, 15, 31, 30, 63, 63, 123, 128, 255, 252, 511, 510, 1015, 1023, 2047, 2040, 4092, 4095, 8176, 8190, 16383, 16365, 32767, 32768, 65503, 65535, 131061, 131040, 262143, 262143, 524223, 524280, 1048575, 1048509, 2097151

Offset: 0

David W. Wilson

Andrew Howroyd, Table of n, a(n) for n = 0..200
J. E. Iglesias, Enumeration of closest-packings by the space group: a simple approach, Z. Krist. 221 (2006) 237-245, Table 3.
Index entries for sequences related to Lyndon words

Cf. A045665, A045674, A045680, A011947 (bisection?).

A045683 := proc(p)
    option remember ;
    if p = 0 then
        return 1;
    end if;
    a := 2^(floor((p+1)/2)-1) ;
    for d in numtheory[divisors](p) do
        if d >1 and type(d,'odd') then
            a := a-procname(p/d) ;
        end if;
    end do:
    a ;
end proc:
seq(A045683(p),p=0..30) ; # [Iglesias eq 12] R. J. Mathar, Apr 15 2024

b[0] = 1; b[n_] := Module[{t = 0, r = n}, While[EvenQ[r], r = Quotient[r, 2]; t += 2^(r-1)]; t + 2^Quotient[r, 2]];
a[0] = 1; a[n_] :=  DivisorSum[n, MoebiusMu[n/#]*b[#]&];
Table[a[n], {n, 0, 43}] (* Jean-François Alcover, Sep 30 2017, after Andrew Howroyd *)

a(n)={if(n<1, n==0, sumdiv(n, d, if(d%2, moebius(d)*2^((n/d-1)\2))))} \\ Andrew Howroyd, Oct 01 2019

Moebius transform of A045674. - Andrew Howroyd, Sep 29 2017
From Andrew Howroyd, Oct 02 2019: (Start)
a(n) = Sum_{d|n, d odd} mu(d) * 2^floor((n/d-1)/2) for n > 0.
G.f.: 1 + Sum_{k>0} mu(2*k-1)*x^(2*k-1)*(1 + x^(2*k-1))/(1 - 2*x^(4*k-2)).
(End)

A045656 Number of 2n-bead balanced binary strings, rotationally equivalent to reverse, complement and reversed complement.

Original entry on oeis.org

1, 2, 6, 8, 22, 32, 48, 100, 150, 260, 336, 684, 784, 1640, 1868, 3728, 4246, 8672, 9372, 19420, 20752, 42736, 45700, 94164, 98832, 204632, 214584, 441764, 460524, 950216, 985968, 2031556, 2101398, 4323888, 4465056, 9174400, 9444988

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000984, A023900, A045665, A045653, A045654, A045655, A045674.

b[n_] := Module[{t = 0, r = n}, If[n == 0, 1,  While[Mod[r, 2] == 0, r = r/2; t += 2^(r - 1)]; t + 2^Quotient[r, 2]]];
c[n_] := Sum[MoebiusMu[d]*d, {d, Divisors[n]}];
a[n_] := If[n == 0, 1, 2*Sum[c[n/d]*d*b[d], {d, Divisors[n]}]];
a /@ Range[0, 36] (* Jean-François Alcover, Sep 23 2019, from PARI *)

\\ here b(n) is A045674,  c(n) is A023900.
b(n) = if(n<1, n==0, my(t=0, r=n); while(r%2==0, r=r/2; t+=2^(r-1)); t + 2^(r\2));
c(n) = {sumdiv(n,d, moebius(d)*d)}
a(n) = if(n<1, n==0, 2*sumdiv(n, d, c(n/d)*d*b(d))); \\ Andrew Howroyd, Sep 15 2019

From Andrew Howroyd, Sep 15 2019: (Start)
Inverse Moebius transform of A045665.
a(n) = 2*Sum_{d|n} A023900(n/d)*d*A045674(d) for n > 0. (End)

A045668 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to complement, inequivalent to reverse and reversed complement.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 24, 28, 128, 252, 720, 1364, 3360, 6552, 14616, 29040, 61440, 122400, 253008, 504868, 1028160, 2054388, 4149288, 8294444, 16679040, 33349800, 66895920, 133775712, 267976800, 535920696, 1072758840, 2145452092

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A011948, A045663, A045665, A045686.

It seems that a(n) = 4*n*A011948(n). - Ralf Stephan, Aug 30 2003
From Andrew Howroyd, Sep 14 2019: (Start)
a(n) = 2*n*A045686(n).
a(n) = A045663(n) - A045665(n). (End)

A045669 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to reversed complement, inequivalent to reverse and complement.

Original entry on oeis.org

0, 0, 0, 12, 32, 120, 288, 784, 1792, 4284, 9600, 21824, 47520, 104832, 225792, 487260, 1040384, 2219520, 4699296, 9942016, 20930400, 43994748, 92184576, 192843776, 402451200, 838655400, 1744404480, 3623423328, 7515275040

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A045664, A045665, A045687.

a(n) = 2*n*A045687(n).

a(n) = A045664(n) - A045665(n). - Andrew Howroyd, Sep 14 2019

A045666 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally inequivalent to reverse, complement and reversed complement.

Original entry on oeis.org

0, 0, 0, 0, 0, 80, 384, 2352, 9856, 42840, 169280, 676720, 2630688, 10265216, 39777248, 154498200, 599556096, 2330826752, 9068386320, 35332969392, 137817005440, 538204062984, 2103970896544, 8233197139552, 32247052083840

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A007727, A045662, A045663, A045664, A045665, A045684.

a(n) = 2*n*A045684(n).

a(n) = A007727(n) - A045662(n) - A045663(n) - A045664(n) + 2*A045665(n). - Andrew Howroyd, Sep 14 2019

A045667 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to reverse, inequivalent to complement and reversed complement.

Original entry on oeis.org

0, 0, 0, 0, 16, 20, 168, 168, 896, 972, 4600, 4840, 20880, 22360, 93744, 99060, 405504, 428876, 1738080, 1828104, 7359600, 7716240, 30982864, 32355664, 129654144, 135002900, 540570160, 561187116, 2245878544, 2325812528

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A045662, A045665, A045685.

a(n) = 2*n*A045685(n).

a(n) = A045662(n) - A045665(n). - Andrew Howroyd, Sep 14 2019

cp's OEIS Frontend

A045683 Number of 2n-bead balanced binary necklaces of fundamental period 2n which are equivalent to their reverse, complement and reversed complement.

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A045656 Number of 2n-bead balanced binary strings, rotationally equivalent to reverse, complement and reversed complement.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A045668 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to complement, inequivalent to reverse and reversed complement.

Views

Author

Keywords

Links

Crossrefs

Formula

A045669 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to reversed complement, inequivalent to reverse and complement.

Views

Author

Keywords

Links

Crossrefs

Formula

A045666 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally inequivalent to reverse, complement and reversed complement.

Views

Author

Keywords

Links

Crossrefs

Formula

A045667 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to reverse, inequivalent to complement and reversed complement.

Views

Author

Keywords

Links

Crossrefs

Formula