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A057079
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Periodic sequence 1,2,1,-1,-2,-1...; expansion of (1+x)/(1-x+x^2).
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36
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1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Inverse binomial transform of A057083. Binomial transform of A061347. The sums of consecutive pairs of elements give A084103. - Paul Barry (pbarry(AT)wit.ie), May 15 2003
Hexaperiodic sequence identical to its third differences. - Paul Curtz (bpcrtz(AT)free.fr), Dec 13 2007
a(n+1) is the Hankel transform of A001700(n+1)-A001700(n). [From Paul Barry (pbarry(AT)wit.ie), Apr 21 2009]
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
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, sqrt(3)); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 1)= A010892(n).
a(n) =2*cos((n-1)*pi/3) =a(n-1)-a(n-2) =-a(n-3) =a(n-6) =(A022003(n+1)+1)*(-1)^[n/3]. Unsigned a(n) =4-a(n-1)-a(n-2) - Henry Bottomley (se16(AT)btinternet.com), Mar 29 2001
a(n)=(-1)^Floor[n/3]+((-1)^Floor[(n-1)/3]+(-1)^Floor[(n+1)/3])/2 - Mario Catalani (mario.catalani(AT)unito.it), Jan 07 2003
a(n)=(1/2-sqrt(3)i/2)^(n-1)+(1/2+sqrt(3)i/2)^(n-1)=cos(pi*n/3)+sqrt(3)sin(pi*n/3) - Paul Barry (pbarry(AT)wit.ie), Mar 15 2004
The period 3 sequence (2, -1, -1, ...) has a(n)=2cos(2pi*n/3)=(-1/2-sqrt(3)i/2)^n+(-1/2+sqrt(3)i/2)^n - Paul Barry (pbarry(AT)wit.ie), Mar 15 2004
Euler transform of length 6 sequence [ 2, -2, -1, 0, 0, 1]. - Michael Somos Jul 14 2006
G.f.: (1+x)/(1-x+x^2) = (1-x^2)^2*(1-x^3)/((1-x)^2*(1-x^6)) . a(2-n)==a(n) . - Michael Somos Jul 14 2006
a(n)=-1/6*{2*(n mod 6)+[(n+1) mod 6]-[(n+2) mod 6]-2*[(n+3) mod 6]-[(n+4) mod 6]+[(n+5) mod 6]} with n>=0 - Paolo P. Lava (paoloplava(AT)gmail.com), Nov 20 2006
a(n)=A033999(A002264(n))*(A000035(A010872(n))+1). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 20 2007
a(n)=(3*A033999(A002264(n))-A033999(n))/2. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 20 2007
a(n)=(-1)^floor(n/3)*((n mod 3) mod 2 + 1). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 20 2007
a(n)=(3*(-1)^floor(n/3)-(-1)^n)/2. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 20 2007
a(n)=(-1)^((n-1)/3)+(-1)^((1-n)/3) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), May 13 2010]
E.g.f.: E(x) = S(0), S(k) = 1 + 2*x/(6*k+1 - x*(6*k+1)/(4*(3*k+1) + x + 4*x*(3*k+1)/(6*k + 3 - x - x*(6*k+3)/(3*k + 2 + x - x*(3*k+2)/(12*k + 10 + x - x*(12*k+10)/(x - (6*k+6)/S(k+1))))))); (continued fraction). - Sergei N. Gladkovskii, Dec 14 2011
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PROG
| (PARI) a(n)=[1, 2, 1, -1, -2, -1][n%6+1] /* Michael Somos Jul 14 2006 */
(PARI) {a(n)=if(n<0, n=2-n); polcoeff((1+x)/(1-x+x^2)+x*O(x^n), n)} /* Michael Somos Jul 14 2006 */
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CROSSREFS
| A049310, A010892. Apart from signs, same as A061347.
Cf. A061347.
a(n)=A010892(n)+A010892(n-1)
Cf. A002264, A010872.
Sequence in context: A100051 A122876 A100063 * A132419 A131556 A107751
Adjacent sequences: A057076 A057077 A057078 * A057080 A057081 A057082
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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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