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 A011655 Period 3: repeat [0, 1, 1]. 119
 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 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)_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 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 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. Index entries for linear recurrences with constant coefficients, signature (0,0,1). 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 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) 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} 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 EXAMPLE 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 + ... MAPLE A011655:=n->(n^2 mod 3): seq(A011655(n), n=0..100); # Wesley Ivan Hurt, Apr 16 2015 MATHEMATICA 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 *) PROG (PARI) {a(n) = sign(n%3)}; (PARI) a(n)=!!(n%3) \\ Jaume Oliver Lafont, Mar 24 2009 (PARI) a(n)=n%3>0 \\ M. F. Hasler, Feb 17 2018 (Haskell) a011655 = fromEnum . ((/= 0) . (`mod` 3)) a011655_list = cycle [0, 1, 1]  -- Reinhard Zumkeller, Apr 07 2012 (Magma) [(n^2 mod 3) : n in [0..100]]; // Wesley Ivan Hurt, Apr 16 2015 (Python) def A011655(n): return int(bool(n%3)) # Chai Wah Wu, May 25 2022 CROSSREFS 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. Sequence in context: A280261 A174784 A092220 * A102283 A128834 A022928 Adjacent sequences:  A011652 A011653 A011654 * A011656 A011657 A011658 KEYWORD nonn,mult,easy AUTHOR EXTENSIONS Better name from Omar E. Pol, Oct 28 2013 STATUS approved

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Last modified September 27 10:36 EDT 2022. Contains 357057 sequences. (Running on oeis4.)