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 A057078 Periodic sequence 1,0,-1,...; expansion of (1+x)/(1+x+x^2). 46
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Partial sums of signed sequence is shifted unsigned one: |a(n+2)|= A011655(n+1). With interpolated zeros, a(n) = sin(5*Pi*n/6 + Pi/3)/sqrt(3) + cos(Pi*n/6 + Pi/6)/sqrt(3); this gives the diagonal sums of the Riordan array (1-x^2, x(1-x^2)). - Paul Barry, Feb 02 2005 From Tom Copeland, Nov 02 2014: (Start) With a shift and a sign change the o.g.f of this array becomes the compositional inverse of the shifted Motzkin or Riordan numbers A005043, (x - x^2) / (1 - x + x^2) = x*(1-x) / (1 - x*(1-x)) = x*(1-x) + [x*(1-x)]^2 + ... . Expanding each term of this series and arranging like powers of x in columns gives skewed rows of the Pascal triangle and reading along the columns gives (mod-signs and indexing) A011973, A169803, and A115139 (see also A091867, A092865, A098925, and A102426 for these term-by-term expansions and A030528). (End) LINKS Ralph E. Griswold, Shaft Sequences Index entries for linear recurrences with constant coefficients, signature (-1,-1). FORMULA a(n) = S(n, -1) + S(n - 1, -1) = S(2*n, 1); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, -1) = A049347(n). S(n, 1)= A010892(n). From Mario Catalani (mario.catalani(AT)unito.it), Jan 08 2003: (Start) a(n) = (1/2)*((-1)^floor(2*n/3) + (-1)^floor((2*n+1)/3)). a(n) = -a(n-1) - a(n-2). a(n) = A061347(n) - A049347(n+2). (End) a(n) = Sum_{k=0..n} binomial(n+k, 2k)*(-1)^(n-k) = Sum_{k=0..floor((n+1)/2)} binomial(n+1-k, k)*(-1)^(n-k). - Mario Catalani (mario.catalani(AT)unito.it), Aug 20 2003 Binomial transform is A010892. a(n) = 2*sqrt(3)*sin(2*Pi*n/3 + Pi/3)/3. - Paul Barry, Sep 13 2003 a(n) = cos(2*Pi*n/3) + sin(2*Pi*n/3)/sqrt(3). - Paul Barry, Oct 27 2004 a(n) = Sum_{k=0..n} (-1)^A010060(2n-2k)*(binomial(2n-k, k) mod 2). - Paul Barry, Dec 11 2004 a(n) = -(1/3)*(2*(n mod 3) - (n+1) mod 3 - (n+2) mod 3). - Paolo P. Lava, Oct 09 2006 a(n) = (4/3)*(|sin(Pi*(n-2)/3)| - |sin(Pi*n/3)|)*|sin(Pi*(n-1)/3)|. - Hieronymus Fischer, Jun 27 2007 a(n) = 1 - (n mod 3) = 1 + 3*floor(n/3)) - n. - Hieronymus Fischer, Jun 27 2007 a(n) = 1 - A010872(n) = 1 + 3*A002264(n) - n. - Hieronymus Fischer, Jun 27 2007 Euler transform of length 3 sequence [0, -1, 1]. - Michael Somos, Oct 15 2008 a(n) = a(n-1)^2 - a(n-2)^2 with a(0) = 1, a(1) = 0. - Francesco Daddi, Aug 02 2011 a(n) = A049347(n) + A049347(n-1). - R. J. Mathar, Jun 26 2013 EXAMPLE 1 - x^2 + x^3 - x^5 + x^6 - x^8 + x^9 - x^11 + x^12 - x^14 + x^15 + ... MAPLE A057078:=n->1-(n mod 3); seq(A057078(n), n=0..100); # Wesley Ivan Hurt, Dec 06 2013 MATHEMATICA a[n_] := {1, 0, -1}[[Mod[n, 3] + 1]] (* Jean-François Alcover, Jul 05 2013 *) CoefficientList[Series[(1 + x) / (1 + x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 03 2014 *) LinearRecurrence[{-1, -1}, {1, 0}, 90] (* Ray Chandler, Sep 15 2015 *) PROG (PARI) {a(n) = [1, 0, -1][n%3 + 1]} /* Michael Somos, Oct 15 2008 */ (Haskell) a057078 = (1 -) . (`mod` 3) -- Reinhard Zumkeller, Mar 22 2013 (Sage) def A057078(): x, y = -1, 0 while True: yield -x x, y = y, -x -y a = A057078(); [next(a) for i in range(40)] # Peter Luschny, Jul 11 2013 CROSSREFS Cf. A049310, A010892, A011655. A049347(n) = a(-n). Cf. A005043, A011973, A169803, A115139, A091867, A092865, A098925, A102426, A030528. Sequence in context: A305385 A260190 A260192 * A204418 A127245 A175192 Adjacent sequences: A057075 A057076 A057077 * A057079 A057080 A057081 KEYWORD easy,sign AUTHOR Wolfdieter Lang, Aug 04 2000 STATUS approved

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Last modified December 7 21:25 EST 2022. Contains 358669 sequences. (Running on oeis4.)