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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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16
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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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The sequences A001607, A077020, A107920, A167433, A169998 are all essentially the same except for signs.
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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T. D. Noe, Table of n, a(n) for n=0..500
Y. Bilu, G. Hanrot, P. M. Voutier and M. Mignotte, Existence of primitive divisors of Lucas and Lehmer numbers, [Research Report] RR-3792, INRIA. 1999, pp. 41, HAL Id : inria-00072867.
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 82.
Erwin Just, Problem E2367, Amer. Math. Monthly, 79 (1972), 772.
G. P. Michon, Never Back to -1.
M. Mignotte, Propriétés arithmétiques des suites récurrentes, Besançon, 1988-1989, see p. 14. In French.
Eric Weisstein's World of Mathematics, Lehmer Number
Index entries for linear recurrences with constant coefficients, signature (-1,-2).
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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
a(n) = (1/7*I)*sqrt(7)*((-1/2-(1/2*i)*sqrt(7))^n - (-1/2+(1/2*i)*sqrt(7))^n), where i=sqrt(-1). - Paolo P. Lava, Jul 19 2011
x/(x^2+x+2) = Sum_{n>=0} a(n)*(x/2)^n. - Benoit Cloitre, Mar 12 2002
4*2^n = A002249(n)^2+7*A001607(n)^2. See A077020, A077021.
a(n+1) = Sum_{k=0..n} A172250(n,k)*(-1)^k. - Philippe Deléham, Feb 15 2012
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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Apart from signs, same as A077020.
Sequence in context: A188509 A265707 A188146 * A167433 A077020 A107920
Adjacent sequences: A001604 A001605 A001606 * A001608 A001609 A001610
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KEYWORD
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sign,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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