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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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12
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| 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
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REFERENCES
| D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966.
Erwin Just, Problem E2367, Amer. Math. Monthly, 79 (1972), 772.
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
| 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
G. P. Michon, Never Back to -1.
Eric Weisstein's World of Mathematics, Lehmer Number
Index entries for sequences related to linear recurrences with constant coefficients, signature (-1,-2).
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FORMULA
| 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 (vladeta(AT)eunet.rs), 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,inf,a(n)*(x/2)^n) - Benoit Cloitre, Mar 12 2002
4*2^n = A002249(n)^2+7*A001607(n)^2. See A077020, A077021.
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MATHEMATICA
| LinearRecurrence[{-1, -2}, {0, 1}, 60] (* From Harvey P. Dale, Aug 21 2011 *)
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PROG
| (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
| Apart from signs, same as A077020.
Sequence in context: A134249 A188509 A188146 * A167433 A077020 A107920
Adjacent sequences: A001604 A001605 A001606 * A001608 A001609 A001610
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KEYWORD
| sign,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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