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%I #28 Feb 23 2018 04:42:01
%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,
%T 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,
%U 1,0,0,0,1,1,1,1,1,1,0,0,0
%N A binary m-sequence: expansion of reciprocal of x^11 + x^2 + 1 (mod 2, shifted by 10 initial 0's).
%C Sequence is 2047-periodic. - _Ray Chandler_, Dec 10 2016
%C 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
%D S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967.
%D H. D. Lueke, Korrelationssignale, Springer 1992, pp. 43-48.
%D F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1978, p. 408.
%H Ray Chandler, <a href="/A011724/b011724.txt">Table of n, a(n) for n = 0..3000</a>
%H <a href="/index/Rec#order_2047">Index entries for linear recurrences with constant coefficients</a>, order 2047.
%F G.f. = x^10/(1+x^2+x^11) over GF(2). - _M. F. Hasler_, Feb 17 2018
%F a(n) == a(n-2) + a(n-11) (mod 2) for n >= 11. - _Robert Israel_, Feb 18 2018
%p for i from 0 to 9 do a[i]:= 0 od: a[10]:= 1:
%p for i from 11 to 200 do a[i]:= a[i-2]+a[i-11] mod 2 od:
%p seq(a[i],i=0..200); # _Robert Israel_, Feb 18 2018
%t Join[Table[0, 10], Mod[CoefficientList[1/(x^11+x^2+1) + O[x]^72, x], 2]] (* _Jean-François Alcover_, Feb 23 2018 *)
%o (PARI) A011724_vec=Vec(lift(Mod(1,2)/(1+x^2+x^11)+O(x^2037)),-2047);
%o A011724(n)=A011724_vec[n%2047+1] \\ Faster than polcoeff(...). - _M. F. Hasler_, Feb 17 2018
%Y Cf. A011655, A011656, ..., A011745 for other binary m-sequences, and A011746..A011751 for similar expansions over GF(2).
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
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