A049347 Period 3: repeat [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

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• sign

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

The binomial transform is 1, 0, -1, -1, 0, 1, 1, 0, -1, -1.. (see A010891). The inverse binom. transform is 1, -2, 3, -3, 0, 9, -27, 54, -81.. (see A057682). - R. J. Mathar, Feb 25 2023

Examples

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

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. Mendès France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012.
• Elena Barcucci, Antonio Bernini, Stefano Bilotta and Renzo Pinzani, Non-overlapping matrices, arXiv:1601.07723 [cs.DM], 2016.
• George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021.
• 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

Crossrefs

Cf. A001006, A010892, A057078, A057083, A102283, A106510, A130716, A143818, A194770.

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

Programs

Magma

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

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 *)
Table[DirichletCharacter[3, 2, n + 1], {n, 0, 29}] (* Steven Foster Clark, May 29 2019 *)
Table[Mod[n + 2, 3] - 1, {n, 0, 20}] (* Michael Somos, Sep 24 2019 *)
Table[ChebyshevU[n, -1/2], {n, 0, 20}] (* Eric W. Weisstein, Jan 09 2024 *)
ChebyshevU[Range[0, 20], -1/2] (* Eric W. Weisstein, Jan 09 2024 *)

Maxima

A049347(n) := block(
        [1,-1,0][1+mod(n,3)]
)$ /* R. J. Mathar, Mar 19 2012 */

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 */

Sage

def A049347():
    x, y = 1, -1
    while True:
        yield x
        x, y = y, -x - y
a = A049347(); [next(a) for i in range(40)] # Peter Luschny, Jul 11 2013

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

From Michael Somos, Oct 03 2006: (Start)

G.f.: (1 - x) /(1 - x^3).

a(n) = -a(1-n) = -a(n-1) - a(n-2) = a(n-3). (End)

From Michael Somos, Apr 29 2012: (Start)

G.f.: 1 / (1 + x / ( 1 - x / (1 + x))).

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). (End)

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

a(n) = A102283(n+1) for all n in Z. - Michael Somos, Sep 24 2019

E.g.f.: exp(-x/2)*(3*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2))/3. - Stefano Spezia, Oct 26 2022

Extensions

Edited by Charles R Greathouse IV, Mar 23 2010