A011746 -id:A011746 - cp's OEIS frontend

A011745 A binary m-sequence: expansion of reciprocal of x^32 + x^28 + x^27 + x + 1 (mod 2, shifted by 31 initial 0's).

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

Offset: 0

N. J. A. Sloane

Periodic with period 2^32-1 = 3*5*17*257*65537 = 4294967295. - M. F. Hasler, Feb 17 2018

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.

Index entries for linear recurrences with constant coefficients, order 4294967295.
Index entries for periodic sequences with large period.

Cf. A011655..A011744 for other binary m-sequences, and A011746..A011751 for similar expansions over GF(2).

Join[Table[0, 31], Mod[CoefficientList[1/(x^32 + x^28 + x^27 + x + 1) + O[x]^50, x], 2]] (* Jean-François Alcover, Feb 23 2018 *)

A011745_vec=concat([1..31]*0,Vec(1/(x^32+x^28+x^27+x+1)+O(x^99))%2)
A=matrix(N=32,N,i,j,if(i>1,i==j+1,setsearch([1,27,28,N],j)))*Mod(1,2);
A011745(n)=lift((A^(n-#A+1))[1,1]) \\ M. F. Hasler, Feb 17 2018

Edited by M. F. Hasler, Feb 17 2018

A088002 Expansion of (1+x^2)/(1+x^2+x^5).

Original entry on oeis.org

1, 0, 0, 0, 0, -1, 0, 1, 0, -1, 1, 1, -2, -1, 3, 0, -4, 2, 5, -5, -5, 9, 3, -14, 2, 19, -11, -22, 25, 20, -44, -9, 66, -16, -86, 60, 95, -126, -79, 212, 19, -307, 107, 386, -319, -405, 626, 298, -1012, 21, 1417, -647, -1715, 1659, 1694, -3076, -1047, 4791, -612, -6485, 3688, 7532, -8479, -6920, 14964, 3232

Offset: 0

N. J. A. Sloane, Nov 02 2003

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,-1,0,0,-1)

Cf. A011746.

f:= gfun:-rectoproc({a(n+5)=-a(n)-a(n+3),a(0)=1,
seq(a(i)=0,i=1..4)},a(n),remember):
map(f, [$1..100]); # Robert Israel, Jul 18 2016

CoefficientList[Series[(1+x^2)/(1+x^2+x^5),{x,0,80}],x] (* or *) LinearRecurrence[ {0,-1,0,0,-1},{1,0,0,0,0},80] (* Harvey P. Dale, Jan 08 2023 *)

Vec((1+x^2)/(1+x^2+x^5)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

A011744 A binary m-sequence: expansion of reciprocal of x^31 + x^3 + 1 (mod 2, shifted by 30 initial 0's).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Periodic with period 2^31-1 = 2147483647. - M. F. Hasler, Feb 17 2018

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.

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

Join[Table[0, 30], Mod[CoefficientList[1/(x^31+x^3+1) + O[x]^52, x], 2]] (* Jean-François Alcover, Feb 23 2018 *)

A011744_vec=concat([1..31]*0,Vec(1/(x^32+x^28+x^27+x+1)+O(x^99))%2)
A=matrix(31,31,i,j,if(i>1,i==j+1,setsearch([3,31],j)>0))*Mod(1,2);
A011744(n)=lift((A^(n-30))[1,1]) \\ M. F. Hasler, Feb 17 2018

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

Original entry on oeis.org

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

N. J. A. Sloane

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

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.

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

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

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

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

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

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

A011727 A binary m-sequence: expansion of reciprocal of x^14 + x^12 + x^11 + x + 1 (mod 2, shifted by 13 initial 0's).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

Sequence is 16383-periodic. - Ray Chandler, Dec 10 2016

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.

Ray Chandler, Table of n, a(n) for n = 0..17000
Index entries for linear recurrences with constant coefficients, order 16383.

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

Mod[#, 2] & /@ CoefficientList[Series[x^13/(x^14 + x^12 + x^11 + x + 1), {x, 0, 105}], x] (* Michael De Vlieger, Feb 21 2018 *)

A011727_vec(N)=Vec(lift(Mod(1,2)/(x^14+x^12+x^11+x+1)+O(x^(N-13))),-N) \\ M. F. Hasler, Feb 17 2018

G.f. = x^13/(x^14 + x^12 + x^11 + x + 1) over GF(2). - M. F. Hasler, Feb 17 2018

A011738 A binary m-sequence: expansion of reciprocal of x^25 + x^3 + 1 (mod 2, shifted by 24 initial 0's).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

Sequence is 2^25-1 = 33554431-periodic. - M. F. Hasler, Feb 17 2018

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.

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, order 33554431.
Index entries for periodic sequences with large period.

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

N:= 200: # to get a(0)..a(N)
A:= Array(0..N):
A[24]:= 1:
for n from 25 to N do A[n]:= A[n-3] + A[n-25] mod 2 od:
convert(A,list); # Robert Israel, Mar 25 2018

A=matrix(N=25,N,i,j, if(i>1, i==j+1, setsearch([3,N],j)>0))*Mod(1, 2); a(n)=lift((A^(n-#A+1))[1,1]) \\ M. F. Hasler, Feb 17 2018

G.f. = x^24/(x^25+x^3+1), over GF(2). - M. F. Hasler, Feb 17 2018

A011743 A binary m-sequence: expansion of reciprocal of x^30 + x^16 + x^15 + x + 1 (mod 2, shifted by 29 initial 0's).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

Periodic with length of period 2^30-1. - M. F. Hasler, Feb 17 2018

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.

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

A011743_vec=concat([1..29]*0, Vec(1/(x^30+x^16+x^15+x+1)+O(x^99))%2)
A=matrix(30, 30, i, j, if(i>1, i==j+1, setsearch([1,15,16,30], j)>0))*Mod(1, 2);
A011743(n)=lift((A^(n-29))[1, 1]) \\ M. F. Hasler, Feb 17 2018

A011731 A binary m-sequence: expansion of reciprocal of x^18 + x^7 + 1 (mod 2, shifted by 17 initial 0's).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

Periodic with period of length 2^18 - 1 = 262143. - M. F. Hasler, Feb 17 2018

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.

Index entries for linear recurrences with constant coefficients, order 262143.

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

A=matrix(N=18,N,i,j,if(i>1,i==j+1,setsearch([7,N],j)>0))*Mod(1,2); a(n)=lift((A^(n-#A+1))[1,1]) \\ M. F. Hasler, Feb 17 2018

G.f. = x^17/(x^18 + x^7 + 1), over GF(2). - M. F. Hasler, Feb 17 2018

Edited by M. F. Hasler, Feb 17 2018

A011732 A binary m-sequence: expansion of reciprocal of x^19 + x^6 + x^5 + x + 1 (mod 2, shifted by 18 initial 0's).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

Sequence is 2^19-1 = 524287-periodic. - M. F. Hasler, Feb 17 2018

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.

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

A=matrix(N=19,N,i,j, if(i>1, i==j+1, setsearch([1,5,6,N], j)>0))*Mod(1, 2); a(n)=lift((A^(n-#A+1))[1,1]) \\ M. F. Hasler, Feb 17 2018

Edited by M. F. Hasler, Feb 17 2018

A011733 A binary m-sequence: expansion of reciprocal of x^20 + x^3 + 1 (mod 2, shifted by 19 initial 0's).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

Sequence is 2^20-1 = 1048575-periodic. - M. F. Hasler, Feb 17 2018

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.

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

A=matrix(N=20,N,i,j, if(i>1, i==j+1, setsearch([3,N], j)>0))*Mod(1, 2); a(n)=lift((A^(n-#A+1))[1, 1]) \\ M. F. Hasler, Feb 17 2018

Edited by M. F. Hasler, Feb 17 2018

cp's OEIS Frontend

A011745 A binary m-sequence: expansion of reciprocal of x^32 + x^28 + x^27 + x + 1 (mod 2, shifted by 31 initial 0's).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A088002 Expansion of (1+x^2)/(1+x^2+x^5).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A011744 A binary m-sequence: expansion of reciprocal of x^31 + x^3 + 1 (mod 2, shifted by 30 initial 0's).

Views

Author

Keywords

Comments

References

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A011727 A binary m-sequence: expansion of reciprocal of x^14 + x^12 + x^11 + x + 1 (mod 2, shifted by 13 initial 0's).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A011738 A binary m-sequence: expansion of reciprocal of x^25 + x^3 + 1 (mod 2, shifted by 24 initial 0's).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

PARI

Formula

A011743 A binary m-sequence: expansion of reciprocal of x^30 + x^16 + x^15 + x + 1 (mod 2, shifted by 29 initial 0's).

Views

Author

Keywords

Comments

References

Crossrefs

Programs

PARI

A011731 A binary m-sequence: expansion of reciprocal of x^18 + x^7 + 1 (mod 2, shifted by 17 initial 0's).