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A087204
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Periodic sequence: 2,1,-1,-2,-1,1,...
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6
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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, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1
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OFFSET
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0,1
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COMMENTS
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Satisfies (a(n))^2 = a(2n) + 2. Shifted differences of itself.
Multiplicative with a(2^e) = -1, a(3^e) = -2, a(p^e) = 1 otherwise. David W. Wilson Jun 12, 2005.
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REFERENCES
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A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 176.
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LINKS
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Table of n, a(n) for n=0..101.
Tanya Khovanova, Recursive Sequences
Wikipedia, Lucas sequence
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index to sequences with linear recurrences with constant coefficients, signature (1,-1).
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FORMULA
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a(n) = a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 1.
G.f.: (2-x)/(1-x+x^2). a(n) = Sum[k>=0, (-1)^k*n/(n-k)*C(n-k, k) ].
a(n) = (1/2) {(-1)^[n/3] + 2(-1)^[(n+1)/3] + (-1)^[(n+2)/3] }.
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, Oct 09 2006
Moebius transform is length 6 sequence [ 1, -2, -3, 0, 0, 6]. - Michael Somos Oct 22 2006
a(n)=a(-n)=-a(n-3). - Michael Somos Oct 22 2006
E.g.f. 2*exp(x/2)*cos(sqrt(3)*x/2) = 2*G(0) where G(k)= 1 + x/(2*(3*k+1) + 2*x*(3*k+1)/( 3*k+2 - x - 2*x*(3*k+2)/(2*x + 3*(k+1)/G(k+1)))); (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Aug 12 2012
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EXAMPLE
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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.
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MATHEMATICA
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PadLeft[{}, 108, {2, 1, -1, -2, -1, 1}] (* From Harvey P. Dale, Sep 11 2011 *)
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PROG
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(PARI) {a(n)=[2, 1, -1, -2, -1, 1][n%6+1]} /* Michael Somos, Oct 22 2006 */
(Sage) [lucas_number2(n, 1, 1) for n in xrange(0, 102)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]
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CROSSREFS
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Essentially the same as A057079 and A100051. Pairwise sums of A010892.
Sequence in context: A131556 A107751 A132367 * A101825 A177702 A131534
Adjacent sequences: A087201 A087202 A087203 * A087205 A087206 A087207
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KEYWORD
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easy,sign,mult
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AUTHOR
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Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Oct 19 2003
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EXTENSIONS
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Edited by Ralf Stephan, Feb 04 2005
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STATUS
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approved
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