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A049347
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Periodic with period 3: repeat [1,-1,0].
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74
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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; internal format)
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OFFSET
| 0,1
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COMMENTS
| (G.f.)^(-1)= cyclotomic(3,x) (cyclotomic polynomial).
Self-convolution yields (-1)^n*A099254(n). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 06 2008
Hankel transform of A099324. [From Paul Barry (pbarry(AT)wit.ie), Aug 10 2009]
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REFERENCES
| A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 175.
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients, signature (-1,-1).
Index entries for sequences related to Chebyshev polynomials.
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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 (pbarry(AT)wit.ie), 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 (paoloplava(AT)gmail.com), Oct 09 2006
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EXAMPLE
| 1 - x + x^3 - x^4 + x^6 - x^7 + x^9 - x^10 + x^12 - x^13 + x^15 + ...
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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 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 24 2009]
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CROSSREFS
| Cf. A010892, A057078.
A057078(n) = a(-n). A106510(n+1) = a(n) unless n=0.
Alternating row sums of A049310 (Chebyshev-S). [From Wolfdieter Lang, Nov 04 2011]
Sequence in context: A011646 A016350 A117441 * A010892 A091338 A016345
Adjacent sequences: A049344 A049345 A049346 * A049348 A049349 A049350
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KEYWORD
| easy,sign,mult
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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EXTENSIONS
| Edited by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Mar 23 2010
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