Search: id:A087204
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%I A087204
%S A087204 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,
%T A087204 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,
%U A087204 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 A087204 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,
%W A087204 -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,
%X A087204 -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
%N A087204 Periodic sequence: 2,1,-1,-2,-1,1,...
%C A087204 Satisfies (a(n))^2 = a(2n) + 2. Shifted differences of itself.
%C A087204 Multiplicative with a(2^e) = -1, a(3^e) = -2, a(p^e) = 1 otherwise. David 
               W. Wilson (davidwwilson(AT)comcast.net) Jun 12, 2005.
%D A087204 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of 
               combinatorial proof, M.A.A. 2003, id. 176.
%H A087204 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A087204 <a href="Sindx_Rea.html#recur1">Index entries for recurrences a(n) = 
               k*a(n - 1) +/- a(n - 2)</a>
%F A087204 a(n) = a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 1.
%F A087204 G.f.: (2-x)/(1-x+x^2). a(n) = Sum[k>=0, (-1)^k*n/(n-k)*C(n-k, k) ].
%F A087204 a(n) = (1/2) {(-1)^[n/3] + 2(-1)^[(n+1)/3] + (-1)^[(n+2)/3] }.
%F A087204 a(n)=-(1/6)*[n mod 6+2*((n+1) mod 6)+(n+2) mod 6-(n+3) mod 6-2*((n+4) 
               mod 6)-(n+5) mod 6] - Paolo P. Lava (ppl(AT)spl.at), Oct 09 2006
%F A087204 Moebius transform is length 6 sequence [ 1, -2, -3, 0, 0, 6]. - Michael 
               Somos Oct 22 2006
%F A087204 a(n)=a(-n)=-a(n-3). - Michael Somos Oct 22 2006
%e A087204 a(2) = -1 = a(1) - a(0) = 1 - 2 = ((1+sqrt(-3))/2)^2 + ((1-sqrt(-3))/
               2)^2 = -1 = -2/4 + 2sqrt(-3)/4 - 2/4 -2 sqrt(-3)/4 = -1.
%o A087204 (PARI) {a(n)=[2, 1, -1, -2, -1, 1][n%6+1]} /* Michael Somos Oct 22 2006 
               */
%o A087204 (Other) sage: [lucas_number2(n,1,1) for n in xrange(0, 102)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]
%Y A087204 Essentially the same as A057079 and A100051. Pairwise sums of A010892.
%Y A087204 Sequence in context: A132367 A101825 A057079 this_sequence A131534 A061347 
               A115579
%Y A087204 Adjacent sequences: A087201 A087202 A087203 this_sequence A087205 A087206 
               A087207
%K A087204 easy,sign,mult
%O A087204 0,1
%A A087204 Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Oct 19 2003
%E A087204 Edited by Ralf Stephan, Feb 04 2005

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