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A011724
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A binary m-sequence: expansion of reciprocal of x^11 + x^2 + 1 (mod 2, shifted by 10 initial 0's).
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1
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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
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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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
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MAPLE
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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:
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MATHEMATICA
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Join[Table[0, 10], Mod[CoefficientList[1/(x^11+x^2+1) + O[x]^72, x], 2]] (* Jean-François Alcover, Feb 23 2018 *)
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PROG
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(PARI) A011724_vec=Vec(lift(Mod(1, 2)/(1+x^2+x^11)+O(x^2037)), -2047);
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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