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A011655
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Periodic sequence 0,1,1,...
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56
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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; internal format)
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OFFSET
| 0,1
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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 (pbarry(AT)wit.ie), 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 (A00045) modulo 2. - Stephen Jordan (sjordan(AT)mit.edu), Sep 10 2007
For n>0: a(n) = A084937(n-1) mod 2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 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. [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), Feb 07 2009]
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 04 2009: (Start)
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,...). (End)
Contribution from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 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. [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 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, 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)] [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 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. - DELEHAM Philippe, Jan 31 2012
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REFERENCES
| Marcia Edson, Scott Lewis and Omer Yayenie, THE K-PERIODIC FIBONACCI SEQUENCE AND AN EXTENDED BINET'S FORMULA, INTEGERS 11 (2011) #A32.
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.
P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, NY, 1991, p. 46. [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), Feb 07 2009]
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LINKS
| 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.
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FORMULA
| G.f.: (x + x^2) / (1 - x^3) = Sum_{n>0} x^n - x^(3*n). a(n) = a(n + 3) = a(-n). a(3*n) = 0.
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 (pbarry(AT)wit.ie), Nov 12 2003
a(n) = 2/3*(1-cos(2Pi*n/3)) - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jan 06 2004
a(n)=1-a(n-1)*a(n-2), a(n)=n for n<2 - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 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 - W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 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 (pbarry(AT)wit.ie), Oct 31 2004
a(n)=mod(A002487(n), 2) - Paul Barry (pbarry(AT)wit.ie), 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 (paoloplava(AT)gmail.com), 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 (paoloplava(AT)gmail.com), Jan 22 2007
a(n)=(4/3)*(|sin(pi*(n-2)/3)|+|sin(pi*(n-1)/3)|)*|sin(pi*n/3)| - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 27 2007
a(n)=((n+1) mod 3 + 1) mod 2 = (1-(-1)^(n-3*floor((n+1)/3)))/2 - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 27 2007
a(n) = 2 - a(n-1) - a(n-2) for n>1 - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 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 (reinhard.zumkeller(AT)gmail.com), Dec 27 2008
Sum_{n=1..infinity} a(n)/n^s = (1-1/3^s)*Riemann_zeta(s), s>1 - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2010
a(n)= floor((4*n-5)/3) mod 2. [From Gary Detlefs (gdetlefs(AT)aol.com), May 15 2011]
a(n)=(a(n-1)-a(n-2))^2 with a(0)=0, a(1)=1. [From Francesco Daddi (francesco.daddi(AT)libero.it), Aug 02 2011]
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EXAMPLE
| x + x^2 + x^4 + x^5 + x^7 + x^8 + x^10 + x^11 + x^13 + x^14 + x^16 + x^17 + ...
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MATHEMATICA
| f[n_] := If[Mod[n, 3] == 0, 0, 1]; Array[f, 105, 0] (* RGWv *)
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PROG
| (PARI) {a(n) = sign(n%3)}
(PARI) a(n)=!!(n%3) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 24 2009]
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CROSSREFS
| Partial sums of A057078 give A011655(n+1). Cf. A049347.
A002487. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 27 2008]
A001590 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 04 2009]
Cf. A168185, A145568, A168184, A168182, A168181, A109720, A097325, A011558, A166486, A000035. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 30 2009]
Sequence in context: A092220 * A102283 A128834 A022928 A000494 A022933
Adjacent sequences: A011652 A011653 A011654 * A011656 A011657 A011658
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KEYWORD
| nonn,mult,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 13 2008
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