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A057078 Periodic sequence 1,0,-1...; expansion of (1+x)/(1+x+x^2). 39
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

Table of n, a(n) for n=0..89.

Ralph E. Griswold, Shaft Sequences

Index entries for sequences related to Chebyshev polynomials.

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).

a(n) = (1/2)((-1)^floor(2n/3) + (-1)^floor((2n+1)/3)). a(n) = -a(n-1) - a(n-2). a(n) = A061347(n) - A049347(n+2). - Mario Catalani (mario.catalani(AT)unito.it), Jan 08 2003

a(n) = sum C(n+k, 2k)(-1)^(n-k), k = 0, .., n = sum C(n+1-k, k)(-1)^(n-k), k = 0, .., floor((n+1)/2). - Mario Catalani (mario.catalani(AT)unito.it), Aug 20 2003

Binomial transform is A010892. a(n) = 2sqrt(3)sin(2Pi*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)*mod(binomial(2n-k, k), 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-Fran├žois 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(); [a.next() 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: A174208 A141687 A204418 * A127245 A175192 A088150

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 June 24 15:07 EDT 2017. Contains 288697 sequences.