A045662 -id:A045662 - cp's OEIS frontend

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

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

A045664 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to reversed complement.

Original entry on oeis.org

1, 2, 4, 18, 48, 150, 324, 882, 1920, 4536, 9900, 22506, 48240, 106470, 227556, 490950, 1044480, 2228190, 4708368, 9961434, 20950800, 44037378, 92229588, 192937938, 402549120, 838860000, 1744617420, 3623864832, 7515733680

Offset: 0

David W. Wilson

nonn

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

Cf. A000740, A001037, A007727, A027375, A045662, A045663.

a[n_] := If[n == 0, 1, 2n Sum[MoebiusMu[n/d] 2^(d-1), {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)*2^(d-1)))} \\ Andrew Howroyd, Sep 14 2019

For n >= 1, a(n) = n*A027375(n) = 2*n*A000740(n) = n^2*A001037(n).

a(n) = 2*n*Sum_{d|n} mu(n/d)*2^(d-1) 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.

A045653 Number of 2n-bead balanced binary strings, rotationally equivalent to reverse.

Original entry on oeis.org

1, 2, 6, 8, 38, 52, 216, 268, 1062, 1232, 4956, 5524, 21848, 24000, 95780, 102808, 410662, 437548, 1748592, 1847524, 7384988, 7759144, 31033404, 32449828, 129774936, 135207552, 540807104, 561629852, 2246432996, 2326762744, 9306937416, 9617286180, 38468731942

Offset: 0

David W. Wilson

nonn

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

Cf. A000984, A023900, A045662, A045654, A045655, A045656.

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

Inverse Moebius transform of A045662. - Andrew Howroyd, Sep 15 2019

Terms a(30) and beyond from Andrew Howroyd, Sep 15 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

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

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A045664 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to reversed 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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PARI

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A045653 Number of 2n-bead balanced binary strings, rotationally equivalent to reverse.

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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

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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