A011751 -id:A011751 - cp's OEIS frontend

A011742 A binary m-sequence: expansion of reciprocal of x^29 + x^2 + 1 (mod 2, shifted by 28 initial 0's).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1

Offset: 0

N. J. A. Sloane

nonn

Periodic with period of length 2^29 - 1 = 536870911. - M. F. Hasler, Feb 17 2018

S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967.
H. D. Lueke, Korrelationssignale, Springer 1992, pp. 43-48.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1978, p. 408.

Cf. A011655, A011656, ..., A011745 for other binary m-sequences, and A011746..A011751 for similar expansions over GF(2).

Join[PadRight[{},28,0],Mod[CoefficientList[Series[1/(x^29+x^2+1),{x,0,100}],x],2]] (* Harvey P. Dale, Sep 27 2024 *)

A=matrix(29,29,i,j,if(i>1,i==j+1,setsearch([2,29],j)>0))*Mod(1,2); a(n)=lift((A^(n-28))[1,1]) \\ M. F. Hasler, Feb 17 2018

Edited by M. F. Hasler, Feb 17 2018

A337442 Number of output sequences from the linear feedback shift register whose feedback polynomial coefficients (excluding the constant term) correspond to the binary representation of n.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 2, 4, 6, 2, 4, 4, 2, 6, 4, 4, 8, 4, 2, 6, 2, 4, 8, 2, 4, 8, 4, 2, 6, 2, 2, 8, 14, 2, 6, 4, 8, 8, 4, 6, 6, 8, 12, 4, 4, 2, 8, 6, 2, 12, 8, 2, 8, 8, 2, 4, 4, 2, 4, 12, 6, 4, 6, 10, 20, 2, 4, 8, 2, 12, 6, 2, 2, 6, 4, 8, 16, 8, 2, 8, 4, 4, 16, 2

Offset: 0

Michael Schwartz, Aug 27 2020

a(n) > 1 for n > 0.
It appears that every term after a(2) is even.
It appears that a(2^n) is greater than each preceding term and is greater than or equal to each term up to a(2^(n+1)).
If a(n) = 2, then the nonzero shift register sequence is an m-sequence.

			For n = 3 = 11 in binary, the polynomial is 1+x+x^2 and the 2 shift register sequences are {00..., 01101...}.
For n = 4 = 100 in binary, the polynomial is 1+x^3 and the 4 shift register sequences are {000..., 001001..., 011011..., 111...}.
For n = 6 = 110 in binary, the polynomial is 1+x^2+x^3 and the 2 shift register sequences are {000..., 0010111001...}.
For n = 10 = 1010 in binary, the polynomial is 1+x^2+x^4 and the 4 shift register sequences are {0000..., 0001010001..., 0011110011..., 0110110...}.
For n = 11 = 1011 in binary, the polynomial in 1+x+x^2+x^4. Using a Fibonacci LSFR, if the current state of the register is 0001, the next input bit is 0+0+1=1, and the next state is 0011. If the current state is 0100, the next input bit is 0+0+0=0, and the next state is 1000. The 4 shift register sequences are {0000..., 00011010001..., 00101110010..., 1111...}.

a(2^n) = A000031(n+1).
A011260 counts how many 2's are in the interval [2^(n-1),(2^n)-1].
a(n) = 2 if and only if 2n+1 is in A091250.
Cf. A100447, A001037, A000016, A000013 (definition 2), A000020, A058947.
Cf. A011655..A011751 for examples of binary m-sequences.

cp's OEIS Frontend

A011742 A binary m-sequence: expansion of reciprocal of x^29 + x^2 + 1 (mod 2, shifted by 28 initial 0's).

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