Search: id:A057079
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%I A057079
%S A057079 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,
%T A057079 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,
%U A057079 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
%V A057079 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,
%W A057079 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,
%X A057079 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
%N A057079 Periodic sequence 1,2,1,-1,-2,-1...; expansion of (1+x)/(1-x+x^2).
%C A057079 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
%C A057079 Hexaperiodic sequence identical to its third differences. - Paul Curtz 
               (bpcrtz(AT)free.fr), Dec 13 2007
%C A057079 a(n+1) is the Hankel transform of A001700(n+1)-A001700(n). [From Paul 
               Barry (pbarry(AT)wit.ie), Apr 21 2009]
%H A057079 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A057079 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A057079 <a href="Sindx_Rea.html#recur1">Index entries for recurrences a(n) = 
               k*a(n - 1) +/- a(n - 2)</a>
%H A057079 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A057079 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).
%F A057079 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
%F A057079 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
%F A057079 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
%F A057079 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
%F A057079 Euler transform of length 6 sequence [ 2, -2, -1, 0, 0, 1]. - Michael 
               Somos Jul 14 2006
%F A057079 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
%F A057079 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 (ppl(AT)spl.at), 
               Nov 20 2006
%F A057079 a(n)=A033999(A002264(n))*(A000035(A010872(n))+1). - Hieronymus Fischer 
               (Hieronymus.Fischer(AT)gmx.de), Jun 20 2007
%F A057079 a(n)=(3*A033999(A002264(n))-A033999(n))/2. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Jun 20 2007
%F A057079 a(n)=(-1)^floor(n/3)*((n mod 3) mod 2 + 1). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Jun 20 2007
%F A057079 a(n)=(3*(-1)^floor(n/3)-(-1)^n)/2. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Jun 20 2007
%o A057079 (PARI) a(n)=[1,2,1,-1,-2,-1][n%6+1] /* Michael Somos Jul 14 2006 */
%o A057079 (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 */
%Y A057079 A049310, A010892. Apart from signs, same as A061347.
%Y A057079 Cf. A061347.
%Y A057079 a(n)=A010892(n)+A010892(n-1)
%Y A057079 Cf. A002264, A010872.
%Y A057079 Sequence in context: A107751 A132367 A101825 this_sequence A087204 A131534 
               A061347
%Y A057079 Adjacent sequences: A057076 A057077 A057078 this_sequence A057080 A057081 
               A057082
%K A057079 easy,sign
%O A057079 0,2
%A A057079 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 04 
               2000

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