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A001607
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a(n) = -a(n-1) - 2*a(n-2).
(Formerly M2225 N0883)
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17
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0, 1, -1, -1, 3, -1, -5, 7, 3, -17, 11, 23, -45, -1, 91, -89, -93, 271, -85, -457, 627, 287, -1541, 967, 2115, -4049, -181, 8279, -7917, -8641, 24475, -7193, -41757, 56143, 27371, -139657, 84915, 194399, -364229, -24569, 753027, -703889
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OFFSET
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0,5
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COMMENTS
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Apart from the sign, this is an example of a sequence of Lehmer numbers. In this case, the two parameters, alpha and beta, are (1 +- i*sqrt(7))/2. Bilu, Hanrot, Voutier and Mignotte show that all terms of a Lehmer sequence a(n) have a primitive factor for n > 30. Note that for this sequence, a(30) = 24475 = 5*5*11*89 has no primitive factors. - T. D. Noe, Oct 29 2003
The sequence a(n) is connected with the sequence A110512 (see Witula's comments to this one). - Roman Witula, Jul 27 2012
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 82.
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FORMULA
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G.f.: x/(1+x+2*x^2).
a(n) = Sum_{k=0..n-1} (-1)^(n-k-1)*binomial(n-k-1, k)*2^k = -2/sqrt(7)*(-sqrt(2))^n*(sin(n*arctan(sqrt(7)))). - Vladeta Jovovic, Feb 05 2003
G.f.: x - 2*x^2 + 2*x^2/(G(0)+1) where G(k) = 1 + x/(1 - x/(x - 1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 16 2012
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MATHEMATICA
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LinearRecurrence[{-1, -2}, {0, 1}, 60] (* Harvey P. Dale, Aug 21 2011 *)
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PROG
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(PARI) a(n)=if(n<0, 0, polcoeff(x/(1+x+2*x^2)+x*O(x^n), n))
(PARI) a(n)=if(n<0, 0, 2*imag(((-1+quadgen(-28))/2)^n))
(Magma) [ n eq 1 select 0 else n eq 2 select 1 else -Self(n-1)-2*Self(n-2): n in [1..50] ]; // Vincenzo Librandi, Aug 22 2011
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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