A011724 A binary m-sequence: expansion of reciprocal of x^11 + x^2 + 1 (mod 2, shifted by 10 initial 0's).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Sequence is 2047-periodic. - Ray Chandler, Dec 10 2016

Expansion of x^10/(x^11+x^2+1) over GF(2). Indeed, 2047 is the smallest k > 0 such that (1-x^k) == 0 (mod 1+x^2+x^11, 2), which means that 1/(1+x^2+x^11) is 2047-periodic over GF(2). It appears somewhat nontrivial that the coefficients of x^2037 through x^2046 of 1/(1+x^2+x^11) are zero (mod 2), which "justifies" the shift by 10 leading zeros. - M. F. Hasler, Feb 16 2018

References

• 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.

Links

• Ray Chandler, Table of n, a(n) for n = 0..3000
• Index entries for linear recurrences with constant coefficients, order 2047.

Crossrefs

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

Programs

Maple

for i from 0 to 9 do a[i]:= 0 od: a[10]:= 1:
for i from 11 to 200 do a[i]:= a[i-2]+a[i-11] mod 2 od:
seq(a[i],i=0..200); # Robert Israel, Feb 18 2018

Mathematica

Join[Table[0, 10], Mod[CoefficientList[1/(x^11+x^2+1) + O[x]^72, x], 2]] (* Jean-François Alcover, Feb 23 2018 *)

PARI

A011724_vec=Vec(lift(Mod(1,2)/(1+x^2+x^11)+O(x^2037)),-2047);
A011724(n)=A011724_vec[n%2047+1] \\ Faster than polcoeff(...). - M. F. Hasler, Feb 17 2018

Formula

G.f. = x^10/(1+x^2+x^11) over GF(2). - M. F. Hasler, Feb 17 2018

a(n) == a(n-2) + a(n-11) (mod 2) for n >= 11. - Robert Israel, Feb 18 2018