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A107920
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Lucas and Lehmer numbers with parameters (1+-sqrt(-7))/2.
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23
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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, 802165, 2209943, 605613, -3814273
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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.
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
Row sums of Riordan array (1/(1+2x^2),x/(1+2x^2)). - Paul Barry, Sep 10 2005
Pisano period lengths: 1, 1, 8, 2, 24, 8, 21, 2, 24, 24, 10, 8,168, 21, 24, 4,144, 24,360, 24,... - R. J. Mathar, Aug 10 2012
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..1000
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
Index to sequences with 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)=a(n-1)-2*a(n-2).
a(n+1) = sum{k=0..n, C((n+k)/2, k)*(-2)^((n-k)/2)*(1+(-1)^(n-k))/2}; a(n+1) = sum{k=0..floor(n/2), C(n-k, k)(-2)^k}; - Paul Barry, Sep 10 2005
a(n+1) = Sum_{k, 0<=k<=n} A109466(n,k)*2^(n-k). - Philippe DELEHAM, Oct 26 2008
a(n) = ((1-i*sqrt(7))^n-(1+i*sqrt(7))^n)*i/(2^n*sqrt(7)), where i=sqrt(-1) - Bruno Berselli, Jul 01 2011
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MAPLE
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a:= n-> (Matrix([[1, 1], [ -2, 0]])^n)[1, 2]: seq (a(n), n=0..45); # Alois P. Heinz, Sep 03 2008
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MATHEMATICA
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Join[{a=0, b=1}, Table[c=b-2*a; a=b; b=c, {n, 100}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 21 2011*)
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PROG
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(PARI) a(n)=if(n<0, 0, imag(quadgen(-7)^n))
(Sage) [lucas_number1(n, 1, +2) for n in xrange(0, 46)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
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CROSSREFS
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A001607(n)=-(-1)^n*a(n).
Cf. A010892.
Sequence in context: A001607 A167433 A077020 * A169998 A171998 A159285
Adjacent sequences: A107917 A107918 A107919 * A107921 A107922 A107923
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KEYWORD
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sign,easy
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AUTHOR
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Michael Somos, May 28 2005
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STATUS
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approved
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