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A001607 a(n) = - a(n-1) - 2a(n-2).
(Formerly M2225 N0883)
+0
10
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; listen)
OFFSET

0,5

COMMENT

x/(x^2+x+2)=sum(n=0,inf,a(n)*(x/2)^n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 12 2002

4*2^n = A002249(n)^2+7*A001607(n)^2. See A077020, A077021.

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 (noe(AT)sspectra.com), Oct 29 2003

REFERENCES

D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966.

Erwin Just, Problem E2367, Amer. Math. Monthly, 79 (1972), 772.

LINKS

Y. Bilu, G. Hanrot, P. M. Voutier and M. Mignotte, Existence of primitive divisors of Lucas and Lehmer numbers

Eric Weisstein's World of Mathematics, Lehmer Number

G. P. Michon, Never Back to -1.

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.yu), Feb 05 2003

PROGRAM

(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))

CROSSREFS

Apart from signs, same as A077020.

Adjacent sequences: A001604 A001605 A001606 this_sequence A001608 A001609 A001610

Sequence in context: A038871 A143524 A134249 this_sequence A077020 A107920 A159285

KEYWORD

sign,easy

AUTHOR

N. J. A. Sloane (njas(AT)research.att.com).

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Last modified July 4 09:27 EDT 2009. Contains 160562 sequences.


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