Search: id:A057078
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%I A057078
%S A057078 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,
%T A057078 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,
%U A057078 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
%V A057078 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,
%W A057078 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,
%X A057078 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
%N A057078 Periodic sequence 1,0,-1...; expansion of (1+x)/(1+x+x^2).
%C A057078 Partial sums of signed sequence is shifted unsigned one: |a(n+2)|= A011655(n+1).
%C A057078 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
%H A057078 Ralph E. Griswold, <a href="http://www.cs.arizona.edu/patterns/sequences.html">
               Shaft Sequences</a>
%H A057078 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A057078 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).
%F A057078 G.f.: (1+x)/(1+x+x^2).
%F A057078 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
%F A057078 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
%F A057078 Binomial transform is A010892. a(n)=2sqrt(3)sin(2pi*n/3+pi/3)/3 - Paul 
               Barry (pbarry(AT)wit.ie), Sep 13 2003
%F A057078 a(n)=cos(2*pi*n/3)+sin(2*pi*n/3)/sqrt(3). - Paul Barry (pbarry(AT)wit.ie), 
               Oct 27 2004
%F A057078 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
%F A057078 a(n) = -(1/3)*[2*(n mod 3)-(n+1) mod 3-(n+2) mod 3] - Paolo P. Lava (ppl(AT)spl.at), 
               Oct 09 2006
%F A057078 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
%F A057078 a(n)=1-(n mod 3)=1+3*floor(n/3))-n. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Jun 27 2007
%F A057078 a(n)=1-A010872(n)=1+3*A002264(n)-n. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Jun 27 2007
%F A057078 Euler transform of length 3 sequence [ 0, -1, 1]. - Michael Somos Oct 
               15 2008
%e A057078 1 - x^2 + x^3 - x^5 + x^6 - x^8 + x^9 - x^11 + x^12 - x^14 + x^15 + ...
%o A057078 (PARI) {a(n) = [1, 0, -1][n%3 + 1]} /* Michael Somos Oct 15 2008 */
%Y A057078 A049310, A010892, A011655.
%Y A057078 A049347(n) = a(-n).
%Y A057078 Sequence in context: A071036 A166946 A141687 this_sequence A127245 A088150 
               A117567
%Y A057078 Adjacent sequences: A057075 A057076 A057077 this_sequence A057079 A057080 
               A057081
%K A057078 easy,sign
%O A057078 0,1
%A A057078 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 04 
               2000

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