A011947 -id:A011947 - 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)

A011768 Number of Barlow packings that repeat after exactly n layers.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 3, 6, 7, 16, 21, 43, 63, 129, 203, 404, 685, 1343, 2385, 4625, 8492, 16409, 30735, 59290, 112530, 217182, 415620, 803076, 1545463, 2990968, 5778267, 11201472, 21702686, 42140890, 81830744, 159139498, 309590883, 602935713, 1174779333, 2290915478, 4469734225, 8726815264

Offset: 1

N. J. A. Sloane and Michael OKeeffe (MOKeeffe(AT)asu.edu)

N. J. A. Sloane, Table of n, a(n) for n = 1..200
Dennis S. Bernstein and Omran Kouba, Counting Colorful Necklaces and Bracelets in Three Colors, arXiv:1901.10703 [math.CO], 2019.
E. Estevez-Rams, C. Azanza-Ricardo, J. Martínez García and B. Aragón-Fernández, On the algebra of binary codes representing closed-packed staking sequences, Acta Cryst. A61 (2005), 201-208.
Ernesto Estevez-Rams, Cristy Azanza Ricardo and Beatriz Aragón Fernández, An alternative expression for counting the number of close-packaged polytypes, Z. Krist. 220 (2005) 592-595, Table 1
T. J. McLarnan, The numbers of polytypes in close-packings and related structures, Zeits. Krist. 155, 269-291 (1981).

Cf. A114438, A371991, A371992.

Cf. A011946, A011947, A011948, A011949, A011950, A011951, A011952, A011953, A011954, A011955, A011956.

with(numtheory); read transforms; M:=200;
A:=proc(N,d) if d mod 3 = 0 then 2^(N/d) else (1/3)*(2^(N/d)+2*cos(Pi*N/d)); fi; end;
E:=proc(N) if N mod 2 = 0 then N*2^(N/2) + add( did(N/2,d)*phi(2*d)*2^(N/(2*d)),d=1..N/2) else (N/3)*(2^((N+1)/2)+2*cos(Pi*(N+1)/2)); fi; end;
PP:=proc(N) (1/(4*N))*(add(did(N,d)*phi(d)*A(N,d), d=1..N)+E(N)); end;
for N from 1 to M do t1[N]:=PP(N); od:
P:=proc(N) local s,d; s:=0; for d from 1 to N do if N mod d = 0 then s:=s+mobius(N/d)*t1[d]; fi; od: s; end; for N from 1 to M do lprint(N,P(N)); od: # N. J. A. Sloane, Aug 10 2006

M = 40;
did[m_, n_] := If[Mod[m, n] == 0, 1, 0];
A[n_, d_] := If[Mod[d, 3] == 0, 2^(n/d), (1/3)(2^(n/d) + 2 Cos[Pi n/d])];
EE[n_] := If[Mod[n, 2] == 0, n 2^(n/2) + Sum[did[n/2, d] EulerPhi[2d]* 2^(n/(2 d)), {d, 1, n/2}], (n/3)(2^((n+1)/2) + 2 Cos[Pi(n+1)/2])];
PP[n_] := PP[n] = (1/(4n))(Sum[did[n, d] EulerPhi[d] A[n, d], {d, 1, n}] + EE[n]);
P[n_] := Module[{s = 0, d}, For[d = 1, d <= n, d++, If[Mod[n, d] == 0, s += MoebiusMu[n/d] PP[d]]]; s];
Array[P, M] (* Jean-François Alcover, Apr 21 2020, from Maple *)

apply( {A011768(n)=A371991(n)+if(n%3, 0, n>3, A371992(n/3), 1)}, [1..42]) \\ M. F. Hasler, May 27 2025

a(n) = (A011946(n/4) + A011947((n-2)/4) + A011948(n/2) + A011949(n/2) + A011950((n+1)/2) + A011951(n/2) + A011952(n/2) + A011953(n)) + (A011954((n-3)/6) + A011955(n/6-1) + A011955(n/6) + A011956(n/3)), where the terms with non-integer indices are set to 0. For n > 3, the two parenthesized terms are resp. A371991(n) and A371992(n/3). - Andrey Zabolotskiy, Feb 14 2024 and May 27 2025

More terms from N. J. A. Sloane, Aug 10 2006

A129629 Nonzero bisection of Moebius transform of A082392.

Original entry on oeis.org

1, 1, 3, 7, 14, 31, 63, 123, 255, 511, 1015, 2047, 4092, 8176, 16383, 32767, 65503, 131061, 262143, 524223, 1048575, 2097151, 4194162, 8388607, 16777208, 33554175, 67108863, 134217693, 268434943, 536870911, 1073741823

Offset: 1

Ralf Stephan, May 31 2007

nonn

Possibly identical to A011947.

A129629_upto(N=100)={ my( d=2*N+3, b=Vec( sum( k=0, exponent(d), (x^2^k)/(1-2*x^2^(k+1)),O(x^d)))); vector( N,i, sumdiv(i*2-1,k, moebius((i*2-1)/k)*b[k])) } \\  M. F. Hasler, May 03 2008, updated May 27 2025

a(n+1) = 2^n-A383182(n). - M. F. Hasler, May 03 2008; sequence number corrected by M. F. Hasler, May 27 2025

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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A011768 Number of Barlow packings that repeat after exactly n layers.

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A129629 Nonzero bisection of Moebius transform of A082392.

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