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Search: id:A001607
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| A001607 |
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a(n) = - a(n-1) - 2a(n-2).
(Formerly M2225 N0883)
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+0
10
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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; listen)
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OFFSET
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0,5
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COMMENT
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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
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REFERENCES
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D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966.
Erwin Just, Problem E2367, Amer. Math. Monthly, 79 (1972), 772.
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LINKS
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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.
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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 (vladeta(AT)Eunet.yu), Feb 05 2003
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PROGRAM
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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))
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CROSSREFS
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Apart from signs, same as A077020.
Sequence in context: A130418 A038871 A134249 this_sequence A077020 A107920 A021080
Adjacent sequences: A001604 A001605 A001606 this_sequence A001608 A001609 A001610
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KEYWORD
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sign,easy
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AUTHOR
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njas
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