A045664 -id:A045664 - cp's OEIS frontend

A045655 Number of 2n-bead balanced binary strings, rotationally equivalent to reversed complement.

1, 2, 6, 20, 54, 152, 348, 884, 1974, 4556, 10056, 22508, 48636, 106472, 228444, 491120, 1046454, 2228192, 4713252, 9961436, 20960904, 44038280, 92252100, 192937940, 402599676, 838860152, 1744723896, 3623869388, 7515962172

Offset: 0

David W. Wilson

nonn

a(n) is the number of ordered pairs (a,b) of length n binary sequences such that a and b are equivalent by rotational symmetry. - Geoffrey Critzer, Dec 31 2011
a(n) is the weighted sum of binary strings of length n by their number of distinct images by rotation. There is a natural correspondence between the first 2^(n-1) sequences (starting with a 0) and the 2^(n-1) starting with a 1 by inversion. There is also an internal correspondance by order inversion. - Olivier Gérard, Jan 01 2011
The number of k-circulant n X n (0,1) matrices, which means the number of n X n binary matrices where rows from the 2nd row on are obtained from the preceding row by a cyclic shift by k columns for some 0 <= k < n.  - R. J. Mathar, Mar 11 2017

			a(2)= 6 because there are 6 such ordered pairs of length 2 binary sequences: (00,00),(11,11),(01,01),(10,10),(01,10),(10,01).
a(3)= 20 because the classes of 3-bit strings are 1*(000), 3*(001,010,100), 3*(011,110,101), 1*(111) = 1 + 9 + 9 + 1.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Chuan Guo, J. Shallit, A. M. Shur, On the Combinatorics of Palindromes and Antipalindromes, arXiv preprint arXiv:1503.09112 [cs.FL], 2015.
V. V. Strok, Circulant matrices and the spectra of de Bruijn graphs, Ukr. Math. J. 44 (11) (1992) 1446-1454.

Cf. A000031 counts the string classes.

Cf. A000984, A023900, A045664, A045653, A045654, A045656.

f[n_] := 2*Plus @@ Table[ Length[ Union[ NestList[ RotateLeft, IntegerDigits[b, 2, n], n - 1]]], {b, 0, 2^(n - 1) - 1}]; f[0] = 1; Array[f, 21, 0] (* Olivier Gérard, Jan 01 2012 *)

c(n)={sumdiv(n,d, moebius(d)*d)} \\ A023900
a(n)={if(n<1, n==0, sumdiv(n, d, c(n/d)*d*2^d))} \\ Andrew Howroyd, Sep 15 2019

For n >= 1, a(n) = Sum_{d|n} A045664(d) = Sum_{d|n} d*A027375(d) = Sum_{d|n} d^2*A001037(d).

a(n) = Sum_{d|n} A023900(n/d)*d*2^d. - Andrew Howroyd, Sep 15 2019

A045662 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to reverse.

Original entry on oeis.org

1, 2, 4, 6, 32, 50, 204, 266, 1024, 1224, 4900, 5522, 21600, 23998, 95508, 102750, 409600, 437546, 1747152, 1847522, 7380000, 7758870, 31027876, 32449826, 129752064, 135207500, 540783100, 561628620, 2246337184, 2326762742

Offset: 0

David W. Wilson

nonn

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

Cf. A007727, A045663, A045664, A045680.

a[n_] := If[n == 0, 1, 2n Sum[MoebiusMu[n/d] Binomial[d - Mod[d, 2], Quotient[d, 2]], {d, Divisors[n]}]];
a /@ Range[0, 30] (* Jean-François Alcover, Sep 23 2019, from PARI *)

a(n) = if(n<1, n==0, 2*n*sumdiv(n, d, moebius(n/d) * binomial(d-d%2, d\2))); \\ Andrew Howroyd, Sep 14 2019

a(n) = 2*n*A045680(n) for n > 0.

a(n) = 2*n*Sum_{d|n} mu(n/d) * binomial(2*floor(d/2), floor(d/2)) for n > 0. - Andrew Howroyd, Sep 14 2019

A045663 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to complement.

Original entry on oeis.org

1, 2, 4, 6, 16, 30, 60, 126, 256, 504, 1020, 2046, 4080, 8190, 16380, 32730, 65536, 131070, 262080, 524286, 1048560, 2097018, 4194300, 8388606, 16776960, 33554400, 67108860, 134217216, 268435440, 536870910, 1073740740, 2147483646

Offset: 0

David W. Wilson

nonn

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

Cf. A000048, A007727, A045662, A045664, A064355.

a[n_] := If[n==0, 1, 2n Total[MoebiusMu[#]*2^(n/#)& /@ Select[Divisors[n], OddQ]]/(2n)];
a /@ Range[0, 31] (* Jean-François Alcover, Sep 23 2019 *)

a(n)={if(n<1, n==0, sumdiv(n, d, if(d%2, moebius(d)*2^(n/d))))} \\ Andrew Howroyd, Sep 14 2019

from sympy import mobius, divisors
def A045663(n): return sum(mobius(d)<>(~n&n-1).bit_length(),generator=True)) if n else 1 # Chai Wah Wu, Jul 22 2024

a(n) = 2*n*A000048(n) = n*A064355(n) for n > 0.

a(n) = Sum{d|n, d odd} mu(d) * 2^(n/d) for n > 0. - Andrew Howroyd, Sep 14 2019

A045665 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to reverse, complement and reversed complement.

Original entry on oeis.org

1, 2, 4, 6, 16, 30, 36, 98, 128, 252, 300, 682, 720, 1638, 1764, 3690, 4096, 8670, 9072, 19418, 20400, 42630, 45012, 94162, 97920, 204600, 212940, 441504, 458640, 950214, 981900, 2031554, 2097152, 4323198, 4456380, 9174270, 9434880

Offset: 0

David W. Wilson

nonn

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

Cf. A007727, A045662, A045663, A045664, A045674, A045683.

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

a(n) = 2*n*A045683(n) for n > 0.

a(n) = n * Sum_{d|n, d odd} mu(d) * 2^ceiling(n/(2*d)) for n > 0.

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

cp's OEIS Frontend

A045655 Number of 2n-bead balanced binary strings, rotationally equivalent to reversed complement.

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A045662 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to reverse.

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A045663 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to complement.

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A045665 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to reverse, complement and reversed complement.

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

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

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Author

Keywords

Links

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