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A011724 A binary m-sequence: expansion of reciprocal of x^11 + x^2 + 1 (mod 2, shifted by 10 initial 0's). 1
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified June 24 08:30 EDT 2021. Contains 345416 sequences. (Running on oeis4.)