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

0,1

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.

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A277151 A277155 A277159 * A037807 A037817 A297039

Adjacent sequences: A011721 A011722 A011723 * A011725 A011726 A011727

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved