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A049347 Period 3: repeat [1, -1, 0]. 89
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

G.f. 1/cyclotomic(3, x) (the third cyclotomic polynomial).

Self-convolution yields (-1)^n*A099254(n). - R. J. Mathar, Apr 06 2008

Hankel transform of A099324. [Paul Barry, Aug 10 2009]

A057083(n) = p(-1)  where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0..n. - Michael Somos, Apr 29 2012

a(n) appears, together with b(n) = A099837(n+3) in the formula 2*exp(2*Pi*n*I/3) = b(n) + a(n)*sqrt(3)*I, n >= 0, with I = sqrt(-1). See A164116 for the case N=5. - Wolfdieter Lang, Feb 27 2014

REFERENCES

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 175.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

J.P. Allouche and M. Mendes France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012.

Elena Barcucci, Antonio Bernini, Stefano Bilotta, Renzo Pinzani, Non-overlapping matrices, arXiv:1601.07723 [cs.DM], 2016.

Index entries for linear recurrences with constant coefficients, signature (-1, -1).

Index entries for sequences related to Chebyshev polynomials.

Index to sequences related to inverse of cyclotomic polynomials

FORMULA

G.f.: 1/(1+x+x^2).

a(n) = +1 if n mod 3 = 0, a(n) = -1 if n mod 3 = 1, else 0.

a(n) = S(n, -1) = U(n, -1/2) (Chebyshev's U(n, x) polynomials.)

a(n) = 2*sqrt(3)*cos(2*Pi*n/3 + Pi/6)/3. - Paul Barry, Mar 15 2004

a(n) = Sum_{k >= 0} (-1)^(n-k)*C(n-k, k).

Given g.f. A(x), then B(x) = x * A(x) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 - v + 2*u*v. - Michael Somos, Oct 03 2006

Euler transform of length 3 sequence [-1, 0, 1]. - Michael Somos, Oct 03 2006

a(n) = b(n+1) where b(n) is multiplicative with b(3^e) = 0^e, b(p^e) = 1 if p == 1 (mod 3), b(p^e) = (-1)^e if p == 2 (mod 3). - Michael Somos, Oct 03 2006

G.f.: (1 - x) /(1 - x^3). a(n) = -a(1-n) = -a(n-1) - a(n-2) = a(n-3). - Michael Somos, Oct 03 2006

a(n) = -(1/3)*[(n mod 3) + ((n+1) mod 3) - 2*((n+2) mod 3)] - Paolo P. Lava, Oct 09 2006

G.f.: 1 / (1 + x / ( 1 - x / (1 + x))). - Michael Somos, Apr 29 2012

a(n) = (-1)^n * A010892(n). a(n) * n! = A194770(n+1). Revert transform of A001006. Convolution inverse of A130716. MOBIUS transform of A002324. EULER transform is A111317. BIN1 transform of itself. STIRLING transform is A143818(n+2). - Michael Somos, Apr 29 2012

a(-n) = A057078(n). a(n) = A106510(n+1) unless n=0. - Michael Somos, Oct 15 2008

G.f. A(x) = 1/(1+x+x^2) = Q(0); Q(k) =  1- x/(1 - x^2/(x^2 - 1 + x/(x - 1 + x^2/(x^2 - 1/Q(k+1))))); (continued fraction 3 kind, 5-step ). - Sergei N. Gladkovskii, Jun 19 2012

a(n) = -1 + floor(67/333*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 03 2013

a(n) = -1 + floor(19/26*3^(n+1)) mod 3. - Hieronymus Fischer, Jan 03 2013

a(n) = ceiling((n-1)/3) - ceiling(n/3) + floor(n/3) - floor((n-1)/3). - Wesley Ivan Hurt, Dec 06 2013

a(n) = n + 1 - 3*floor((n+2)/3). - Mircea Merca, Feb 04 2014

EXAMPLE

1 - x + x^3 - x^4 + x^6 - x^7 + x^9 - x^10 + x^12 - x^13 + x^15 + ...

MAPLE

A049347 := proc(n)

    op(modp(n, 3)+1, [1, -1, 0]) ;

end proc:

seq(A049347(n), n=0..100) ; # R. J. Mathar, Aug 06 2016

MATHEMATICA

Flatten[Table[{1, -1, 0}, {27}]] (* Alonso del Arte, Feb 07 2013 *)

CoefficientList[Series[1/Cyclotomic[3, x], {x, 0, 100}], x] (* Vincenzo Librandi, Apr 03 2014 *)

LinearRecurrence[{-1, -1}, {1, -1}, 90] (* Ray Chandler, Sep 15 2015 *)

PROG

(PARI) {a(n) = n++; kronecker( -3, n)} /* Michael Somos, Oct 03 2006 */

(PARI) {a(n) = [1, -1, 0][n%3 + 1]} /* Michael Somos, Oct 15 2008 */

(PARI) a(n)=(n+2)%3-1 /* Jaume Oliver Lafont, Mar 24 2009 */

(Maxima) A049347(n) := block(

        [1, -1, 0][1+mod(n, 3)]

)$ /* R. J. Mathar, Mar 19 2012 */

(Sage)

def A049347():

    x, y = 1, -1

    while true:

        yield x

        x, y = y, -x - y

a = A049347(); [a.next() for i in range(40)] # Peter Luschny, Jul 11 2013

(MAGMA) &cat[[1, -1, 0]: n in [0..90]]; // Vincenzo Librandi, Apr 03 2014

CROSSREFS

Cf. A001006, A010892, A057078, A057083, A106510, A130716, A143818, A194770.

Alternating row sums of A049310 (Chebyshev-S). [Wolfdieter Lang, Nov 04 2011]

Sequence in context: A011646 A016350 A117441 * A010892 A091338 A016345

Adjacent sequences:  A049344 A049345 A049346 * A049348 A049349 A049350

KEYWORD

easy,sign

AUTHOR

Wolfdieter Lang

EXTENSIONS

Edited by Charles R Greathouse IV, Mar 23 2010

STATUS

approved

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Last modified November 20 00:42 EST 2017. Contains 294957 sequences.