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A011655 Period 3: repeat 0, 1, 1. 66
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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

Contribution from Reinhard Zumkeller, Nov 30 2009: (Start)

a(n) = 1-A079978(n); characteristic function of numbers coprime to 3; a(A001651(n)) = 1; a(A008585(n)) = 0;

A000212(n) = SUM(a(k)*(n-k): 0 <= k <= n). (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. - W. 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)_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

REFERENCES

S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967.

L. B. W. Jolley, Summation of Series, Dover (1961).

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.

P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, NY, 1991, p. 46. [Rick L. Shepherd, Feb 07 2009]

LINKS

Table of n, a(n) for n=0..104.

Marcia Edson, Scott Lewis and Omer Yayenie, The k-periodic Fibonacci sequence and an extended Binet's formula, INTEGERS 11 (2011) #A32.

Index entries for two-way infinite sequences

Index entries for sequences related to linear recurrences with constant coefficients, signature (0,0,1).

Index entries for characteristic functions

Index entries for sequences related to Chebyshev polynomials.

FORMULA

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.

a(n) = (1/2)*( (-1)^(floor((2n + 4)/3)) + 1 ). - Mario Catalani (mario.catalani(AT)unito.it), Oct 22 2003

a(n) = mod(Fib(n), 2). - Paul Barry, Nov 12 2003

a(n) = 2/3*(1-cos(2Pi*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) = mod(A002487(n), 2). - Paul Barry, Jan 14 2005

a(n)= n^2 mod 3. a(n) = (1/3)*(2-(r^n+r^(2*n))) where r = (-1+sqrt(-3))/2.  - Bruce Corrigan (scentman(AT)myfamily.com), Aug 08 2005

Contribution 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)

a(n) = {(2/3)*[cos(2*n*Pi/3)+1/2]-1}^2. - Paolo P. Lava, Oct 09 2006

a(n) = (1/9)*{5*(n mod 3)+2*[(n+1) mod 3]-[(n+2) mod 3]} with n >= 0. - Paolo P. Lava, Jan 22 2007

a(n) = (4/3)*(|sin(pi*(n-2)/3)|+|sin(pi*(n-1)/3)|)*|sin(pi*n/3)|. - Hieronymus Fischer, Jun 27 2007

a(n) = ((n+1) mod 3 + 1) mod 2 = (1-(-1)^(n-3*floor((n+1)/3)))/2. - Hieronymus Fischer, Jun 27 2007

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..infinity} 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

G.f.: Q(0) where Q(k) =  1 - x^(2*k)/(1 - x/(x - x^(2*k)/Q(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) = signum(n mod 3). [Wesley Ivan Hurt, Jun 22 2013]

a(n) = A000035(A000032(n)) = A000035(A000045(n)). - Omar E. Pol, Oct 28 2013

EXAMPLE

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

MATHEMATICA

f[n_] := If[Mod[n, 3] == 0, 0, 1];  Array[f, 105, 0] (* Robert G. Wilson v *)

PROG

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

(PARI) a(n)=!!(n%3) [Jaume Oliver Lafont, Mar 24 2009]

(Haskell)

a011655 = fromEnum . ((/= 0) . (`mod` 3))

a011655_list = cycle [0, 1, 1]  -- Reinhard Zumkeller, Apr 07 2012

CROSSREFS

Partial sums of A057078 give A011655(n+1). Cf. 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).

Sequence in context: A092220 * A102283 A128834 A022928 A000494 A022933

Adjacent sequences:  A011652 A011653 A011654 * A011656 A011657 A011658

KEYWORD

nonn,mult,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Reinhard Zumkeller, Apr 13 2008

Better name from Omar E. Pol, Oct 28 2013

STATUS

approved

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Last modified August 31 04:27 EDT 2014. Contains 246235 sequences.