A045654
Number of 2n-bead balanced binary strings, rotationally equivalent to complement.
Original entry on oeis.org
1, 2, 6, 8, 22, 32, 72, 128, 278, 512, 1056, 2048, 4168, 8192, 16512, 32768, 65814, 131072, 262656, 524288, 1049632, 2097152, 4196352, 8388608, 16781384, 33554432, 67117056, 134217728, 268451968, 536870912, 1073774592, 2147483648, 4295033110, 8589934592
Offset: 0
From _Andrew Howroyd_, Jul 06 2025: (Start)
The a(1) = 2 length 2 balanced binary strings are: 01, 10.
The a(2) = 6 strings are: 0101, 1010, 0011, 0110, 1100, 1001.
The a(3) = 8 strings are: 010101, 101010, 000111, 001110, 011100, 111000, 110001, 100011. (End)
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a:= proc(n) option remember;
2^n+`if`(n::even and n>0, a(n/2), 0)
end:
seq(a(n), n=0..33); # Alois P. Heinz, Jul 01 2025
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a(n)={if(n==0, 1, my(s=0); while(n%2==0, s+=2^n; n/=2); s + 2^n)} \\ Andrew Howroyd, Sep 22 2019
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def A045654(n): return sum(1<<(n>>k) for k in range((~n & n-1).bit_length()+1)) if n else 1 # Chai Wah Wu, Jul 22 2024
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
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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 *)
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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
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
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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 *)
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a(n)={if(n<1, n==0, 2*n*sumdiv(n, d, moebius(n/d)*2^(d-1)))} \\ 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
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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
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
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
Showing 1-6 of 6 results.