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A107920 Lucas and Lehmer numbers with parameters (1+-sqrt(-7))/2. 27
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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

This is the Lucas Sequence U_n(P, Q) = U_n(1, 2). V_n(1, 2) = A002249(n). - Raphie Frank, Dec 25, 2013

LINKS

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

Wikipedia, Lucas Sequence

Index to sequences with linear recurrences with constant coefficients, signature (1,-2).

FORMULA

G.f.: x/(1-x+2*x^2).

a(n) = a(n-1) - 2*a(n-2).

a(n) = -(-1)^n*A001607(n).

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 Deléham, 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

(a(2*(A060728(n)) - 4))^2 = (A002249(A060728(n) - 2))^2 = 2^(A060728(n)) - 7 = A227078(n), the Ramanujan-Nagell squares. - Raphie Frank, Dec 25, 2013

MAPLE

a:= n-> (Matrix([[1, 1], [ -2, 0]])^n)[1, 2]: seq (a(n), n=0..45); # Alois P. Heinz, Sep 03 2008

MATHEMATICA

Join[{a=0, b=1}, Table[c=b-2*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011*)

PROG

(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, Apr 22 2009]

CROSSREFS

Sequence in context: A001607 A167433 A077020 * A169998 A171998 A159285

Adjacent sequences:  A107917 A107918 A107919 * A107921 A107922 A107923

KEYWORD

sign,easy

AUTHOR

Michael Somos, May 28 2005

STATUS

approved

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Last modified October 2 12:40 EDT 2014. Contains 247541 sequences.