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A057078
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Periodic sequence 1,0,-1...; expansion of (1+x)/(1+x+x^2).
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23
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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; internal format)
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OFFSET
| 0,1
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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 (pbarry(AT)wit.ie), Feb 02 2005
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LINKS
| Ralph E. Griswold, Shaft Sequences
Index entries for sequences related to Chebyshev polynomials.
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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).
G.f.: (1+x)/(1+x+x^2).
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 (pbarry(AT)wit.ie), Sep 13 2003
a(n)=cos(2*pi*n/3)+sin(2*pi*n/3)/sqrt(3). - Paul Barry (pbarry(AT)wit.ie), Oct 27 2004
a(n)=sum{k=0..n, (-1)^A010060(2n-2k)*mod(binomial(2n-k, k), 2)} - Paul Barry (pbarry(AT)wit.ie), Dec 11 2004
a(n) = -(1/3)*[2*(n mod 3)-(n+1) mod 3-(n+2) mod 3] - Paolo P. Lava (paoloplava(AT)gmail.com), Oct 09 2006
a(n)=(4/3)*(|sin(pi*(n-2)/3)|-|sin(pi*n/3)|)*|sin(pi*(n-1)/3)|. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 27 2007
a(n)=1-(n mod 3)=1+3*floor(n/3))-n. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 27 2007
a(n)=1-A010872(n)=1+3*A002264(n)-n. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 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. [From Francesco Daddi, Aug 02 2011]
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EXAMPLE
| 1 - x^2 + x^3 - x^5 + x^6 - x^8 + x^9 - x^11 + x^12 - x^14 + x^15 + ...
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PROG
| (PARI) {a(n) = [1, 0, -1][n%3 + 1]} /* Michael Somos Oct 15 2008 */
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CROSSREFS
| A049310, A010892, A011655.
A049347(n) = a(-n).
Sequence in context: A174208 A141687 A204418 * A127245 A175192 A088150
Adjacent sequences: A057075 A057076 A057077 * A057079 A057080 A057081
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KEYWORD
| easy,sign
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 04 2000
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