A011751 -id:A011751 - cp's OEIS frontend

A011655 Period 3: repeat [0, 1, 1].

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

Offset: 0

N. J. A. Sloane

A binary m-sequence: expansion of reciprocal of x^2+x+1 (mod 2).
A Chebyshev transform of the Jacobsthal numbers A001045: if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))*A(x/(1+x^2)). - Paul Barry, Feb 16 2004
This is the r = 1 member of the r-family of sequences S_r(n) defined in A092184 where more information can be found.
This is the Fibonacci sequence (A000045) modulo 2. - Stephen Jordan (sjordan(AT)mit.edu), Sep 10 2007
For n > 0: a(n) = A084937(n-1) mod 2. - Reinhard Zumkeller, Dec 16 2007
This is also the Lucas numbers (A000032) mod 2. In general, this is the parity of any Lucas sequence associated with any pair (P,Q) when P and Q are odd; i.e., a(n) = U_n(P,Q) mod 2 = V_n(P,Q) mod 2. See Ribenboim. - Rick L. Shepherd, Feb 07 2009
Starting with offset 1: (1, 1, 0, 1, 1, 0, ...) = INVERTi transform of the tribonacci sequence A001590 starting (1, 2, 3, 6, 11, 20, 37, ...). - Gary W. Adamson, May 04 2009
From Reinhard Zumkeller, Nov 30 2009: (Start)
Characteristic function of numbers coprime to 3.
a(n) = 1 - A079978(n); a(A001651(n)) = 1; a(A008585(n)) = 0;
A000212(n) = Sum_{k=0..n} a(k)*(n-k). (End)
Sum_{k>0} a(k)/k/2^k = log(7)/3. - Jaume Oliver Lafont, Jun 01 2010
The sequence is the principal Dirichlet character of the reduced residue system mod 3 (the other is A102283). Associated Dirichlet L-functions are L(2,chi) = Sum_{n>=1} a(n)/n^2 = 4*Pi^2/27 = A214549, and L(3,chi) = Sum_{n>=1} a(n)/n^3 = 1.157536... = -(psi''(1/3) + psi''(2/3))/54 where psi'' is the tetragamma function. [Jolley eq 309 and arXiv:1008.2547, L(m = 3, r = 1, s)]. - R. J. Mathar, Jul 15 2010
a(n+1), n >= 0, is the sequence of the row sums of the Riordan triangle A158454. - Wolfdieter Lang, Dec 18 2010
Removing the first two elements and keeping the offset at 0, this is a periodic sequence (1, 0, 1, 1, 0, 1, ...). Its INVERTi transform is (1, -1, 2, -2, 2, -2, ...) with period (2,-2). - Gary W. Adamson, Jan 21 2011
Column k = 1 of triangle in A198295. - Philippe Deléham, Jan 31 2012
The set of natural numbers, A000027: (1, 2, 3, ...); is the INVERT transform of the signed periodic sequence (1, 1, 0, -1, -1, 0, 1, 1, 0, ...). - Gary W. Adamson, Apr 28 2013
Any integer sequence s(n) = |s(n-1) - s(n-2)| (equivalently, max(s(n-1), s(n-2)) - min(s(n-1), s(n-2))) for n > i + 1 with s(i) = j and s(i+1) = k, where j and k are not both 0, is or eventually becomes a multiple of this sequence, namely, the sequence repeat gcd(j, k), gcd(j, k), 0 (at some offset). In particular, if j and k are coprime, then s(n) is or eventually becomes this sequence (see, e.g., A110044). - Rick L. Shepherd, Jan 21 2014
For n >= 1, a(n) is also the characteristic function for rational g-adic integers (+n/3)A001651).%20See%20the%20definition%20in%20the%20Mahler%20reference,%20p.%207%20and%20also%20p.%2010.%20-%20_Wolfdieter%20Lang">g and also (-n/3)_g for all integers g >= 2 without a factor 3 (A001651). See the definition in the Mahler reference, p. 7 and also p. 10. - _Wolfdieter Lang, Jul 11 2014
Characteristic function for A007908(n+1) being divisible by 3. a(n) = bit flipped A007908(n+1) (mod 3) = bit flipped A079978(n). - Wolfdieter Lang, Jun 12 2017
Also Jacobi or Kronecker symbol (n/9) (or (n/9^e) for all e >= 1). - Jianing Song, Jul 09 2018
The binomial trans. is 0, 1, 3, 6, 11, 21, 42, 85, 171, 342,.. (see A024495). - R. J. Mathar, Feb 25 2023

			G.f. = x + x^2 + x^4 + x^5 + x^7 + x^8 + x^10 + x^11 + x^13 + x^14 + x^16 + x^17 + ...

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.
K. Mahler, p-adic numbers and their functions, 2nd ed., Cambridge University press, 1981.
Paulo Ribenboim, The Little Book of Big Primes. Springer-Verlag, NY, 1991, p. 46. [Rick L. Shepherd, Feb 07 2009]

Andrei Asinowski, Cyril Banderier and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Marcia Edson, Scott Lewis and Omer Yayenie, The k-periodic Fibonacci sequence and an extended Binet's formula, INTEGERS 11 (2011) #A32.
Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
L. B. W. Jolley, Summation of Series, Dover, (1961).
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,0,1).
Index entries for characteristic functions.
Index entries for sequences related to Chebyshev polynomials..

Partial sums of A057078 give A011655(n+1).
Cf. A035191 (Mobius transform), A001590, A002487, A049347.
Cf. A000035, A011558, A097325, A109720, A145568, A166486, A168181, A168182, A168184, A168185. - Reinhard Zumkeller, Nov 30 2009
Cf. A000027, A000045, A004523 (partial sums), A057078 (first differences).
Cf. A007908, A079978 (bit flipped).
Cf. A011656 - A011751 for other binary m-sequences.
Cf. A002264.

a011655 = fromEnum . ((/= 0) . (`mod` 3))
a011655_list = cycle [0,1,1]  -- Reinhard Zumkeller, Apr 07 2012

[(n^2 mod 3) : n in [0..100]]; // Wesley Ivan Hurt, Apr 16 2015

A011655:=n->(n^2 mod 3): seq(A011655(n), n=0..100); # Wesley Ivan Hurt, Apr 16 2015

A011655[n_] := If[Mod[n, 3] == 0, 0, 1];  Array[A011655, 105, 0] (* Robert G. Wilson v *)
Mod[Fibonacci[Range[0, 99]], 2] (* Alonso del Arte, Jul 20 2017 *)

PARI
```
{a(n) = sign(n%3)};
```

a(n)=!!(n%3) \\ Jaume Oliver Lafont, Mar 24 2009

a(n)=n%3>0 \\ M. F. Hasler, Feb 17 2018

def A011655(n): return int(bool(n%3)) # Chai Wah Wu, May 25 2022

G.f.: (x + x^2) / (1 - x^3) = Sum_{k>0} (x^k - x^(3*k)).
G.f.: x / (1 - x / (1 + x / (1 + x / (1 - 2*x / (1 + x))))). - Michael Somos, Apr 02 2012
a(n) = a(n+3) = a(-n), a(3*n) = 0, a(3*n + 1) = a(3*n + 2) = 1 for all n in Z.
a(n) = (1/2)*( (-1)^(floor((2n + 4)/3)) + 1 ). - Mario Catalani (mario.catalani(AT)unito.it), Oct 22 2003
a(n) = Fibonacci(n) mod 2. - Paul Barry, Nov 12 2003
a(n) = (2/3)*(1 - cos(2*Pi*n/3)). - Ralf Stephan, Jan 06 2004
a(n) = 1 - a(n-1)*a(n-2), a(n) = n for n < 2. - Reinhard Zumkeller, Feb 28 2004
a(n) = 2*(1 - T(n, -1/2))/3 with Chebyshev's polynomials T(n, x) of the first kind; see A053120. - Wolfdieter Lang, Oct 18 2004
a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*A001045(n-2k)/(n-k). - Paul Barry, Oct 31 2004
a(n) = A002487(n) mod 2. - Paul Barry, Jan 14 2005
From Bruce Corrigan (scentman(AT)myfamily.com), Aug 08 2005: (Start)
a(n) = n^2 mod 3.
a(n) = (1/3)*(2 - (r^n + r^(2*n))) where r = (-1 + sqrt(-3))/2. (End)
From Michael Somos, Sep 23 2005: (Start)
Euler transform of length 3 sequence [ 1, -1, 1].
Moebius transform is length 3 sequence [ 1, 0, -1].
Multiplicative with a(3^e) = 0^e, a(p^e) = 1 otherwise. (End)
From Hieronymus Fischer, Jun 27 2007: (Start)
a(n) = (4/3)*(|sin(Pi*(n-2)/3)| + |sin(Pi*(n-1)/3)|)*|sin(Pi*n/3)|.
a(n) = ((n+1) mod 3 + 1) mod 2 = (1 - (-1)^(n - 3*floor((n+1)/3)))/2. (End)
a(n) = 2 - a(n-1) - a(n-2) for n > 1. - Reinhard Zumkeller, Apr 13 2008
a(2*n+1) = a(n+1) XOR a(n), a(2*n) = a(n), a(1) = 1, a(0) = 0. - Reinhard Zumkeller, Dec 27 2008
Sum_{n>=1} a(n)/n^s = (1-1/3^s)*Riemann_zeta(s), s > 1. - R. J. Mathar, Jul 31 2010
a(n) = floor((4*n-5)/3) mod 2. - Gary Detlefs, May 15 2011
a(n) = (a(n-1) - a(n-2))^2 with a(0) = 0, a(1) = 1. - Francesco Daddi, Aug 02 2011
Convolution of A040000 with A049347. - R. J. Mathar, Jul 21 2012
G.f.: Sum_{k>0} x^A001651(k). - L. Edson Jeffery, Dec 05 2012
G.f.: x/(G(0) - x^2) where G(k) = 1 - x/(x + 1/(1 - x/G(k+1))); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 15 2013
For the general case: The characteristic function of numbers that are not multiples of m is a(n) = floor((n-1)/m) - floor(n/m) + 1, with m,n > 0. - Boris Putievskiy, May 08 2013
a(n) = sign(n mod 3). - Wesley Ivan Hurt, Jun 22 2013
a(n) = A000035(A000032(n)) = A000035(A000045(n)). - Omar E. Pol, Oct 28 2013
a(n) = (-n mod 3)^((n-1) mod 3). - Wesley Ivan Hurt, Apr 16 2015
a(n) = (2/3) * (1 - sin((Pi/6) * (4*n + 3))) for n >= 0. - Werner Schulte, Jul 20 2017
a(n) = a(n-1) XOR a(n-2) with a(0) = 0, a(1) = 1. - Chunqing Liu, Dec 18 2022
a(n) = floor((n+2)/3) - floor(n/3) = A002264(n+2) - A002264(n). - Aaron J Grech, Jul 30 2024
E.g.f.: 2*(exp(x) - exp(-x/2)*cos(sqrt(3)*x/2))/3. - Stefano Spezia, Mar 30 2025
Dirichlet g.f.: zeta(s)*(1-1/3^s). - R. J. Mathar, Aug 10 2025

Better name from Omar E. Pol, Oct 28 2013

A011746 Expansion of (1 + x^2)/(1 + x^2 + x^5) mod 2.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Michael Gilleland, Some Self-Similar Integer Sequences
R. Gold, Characteristic linear sequences and their coset functions, J. SIAM Applied. Math., 14 (1966), 980-985.
Index entries for linear recurrences with constant coefficients, signature (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).

Cf. A011662.

Cf. A011747..A011751 for similar sequences and A011655 - A011745 for other binary m-sequences.

series((1+x^2)/(1+x^2+x^5),x,100) mod 2;

Mod[CoefficientList[Series[(1+x^2)/(1+x^2+x^5),{x,0,80}],x],2] (* Harvey P. Dale, Jul 19 2023 *)

A011746_vec=Vec((1+x^2)/(1+x^2+x^5)+O(x^31))%2 \\ For illustrative purpose.
A011746(n)=bittest(377253537,n%31) \\ M. F. Hasler, Feb 17 2018

a(n) = A088002(n) mod 2. - R. J. Mathar, May 26 2008
G.f.: (1+x^5+x^7+x^9+x^10+x^11+x^13+x^14+x^18+x^19+x^20+x^21+x^22+x^25+x^26+x^28)/(1-x^31). - Robert Israel, May 06 2018
a(n) = A011662(n-1). - R. J. Mathar, Jan 12 2024

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).

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, 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

A011656 A binary m-sequence: expansion of reciprocal of x^3 + x^2 + 1 (mod 2), shifted by 2 initial 0's.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Period 7.

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, signature (0,0,0,0,0,0,1).

Cf. A077962.

Cf. A011655..A011751 for other binary m-sequences.

PadLeft[ Mod[ CoefficientList[ Series[1/(1 + x^2 + x^3), {x, 0, 102}], x], 2], 105] (* Robert G. Wilson v *)

A011656_vec(N)=concat([0,0],Vec(lift(O(x^(N-1))+Mod(1,2)/(1+x^2+x^3))))
A011656(n)=(n%7>3)||(n%7==2) \\ Faster than polcoeff(.../(1+x^2+x^3),n-2). - M. F. Hasler, Feb 17 2018

G.f.: (x^6 + x^5 + x^4 + x^2)/(1-x^7). a(n+7) = a(n). - Ralf Stephan, Aug 05 2013

G.f.: x^2/(1 + x^2 + x^3) in GF(2). - M. F. Hasler, Feb 16 2018

A011657 A binary m-sequence: expansion of reciprocal of x^3 + x + 1 (mod 2, shifted by 2 initial 0's).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Period 7.

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, signature (0,0,0,0,0,0,1).

Cf. A011655..A011751 for other binary m-sequences.

Join[{0, 0}, Mod[CoefficientList[1/(x^3 + x + 1) + O[x]^80, x], 2]] (* Jean-François Alcover, Feb 23 2018 *)

A011657(n)=bittest(92,n%7) \\ M. F. Hasler, Feb 17 2018

G.f.: (x^6 + x^3 + x + 1)/(1-x^7), a(n+7) = a(n). - Ralf Stephan, Aug 05 2013

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

cp's OEIS Frontend

A011655 Period 3: repeat [0, 1, 1].

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A011746 Expansion of (1 + x^2)/(1 + x^2 + x^5) mod 2.

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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).

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A011656 A binary m-sequence: expansion of reciprocal of x^3 + x^2 + 1 (mod 2), shifted by 2 initial 0's.

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A011657 A binary m-sequence: expansion of reciprocal of x^3 + x + 1 (mod 2, shifted by 2 initial 0's).

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

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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).

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