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A107920 Lucas and Lehmer numbers with parameters (1+-sqrt(-7))/2. 28
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

Note that (A002249(n)/2)^2 + 7*(a(n)/2)^2 = 2^n for all n in N. This is a specific case of the Lucas sequence identity (V_n/2)^2 - D*(U_n/2)^2 = Q^n where V_n = (a^n + b^n), U_n = (a^n - b^n)/(a - b), Q = (a*b) = 2 and D = (a - b)^2 = -7; a = (1 + sqrt(-7))/2 and b =(1 - sqrt(-7))/2. - Raphie Frank, Nov 26 2015

For the special case where |a(n)| = 1, true for n if and only if n is in {1, 2, 3, 5, 13} = {A215795(n) + 1} = {A060728(n) - 2}, then, additionally, by the Lucas sequence identity (U_2n = U_n*V_n), we have (a(2n)/2)^2 + 7*(a(n)/2)^2 = 2^n. - Raphie Frank, Nov 26 2015

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, [Research Report] RR-3792, INRIA. 1999, pp.41, HAL Id : inria-00072867.

Eric Weisstein's World of Mathematics, Lehmer Number

Wikipedia, Lucas Sequence

Index entries for 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).

From Paul Barry, Sep 10 2005: (Start)

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. (End)

a(n+1) = Sum_{k=0..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 *)

LinearRecurrence[{1, -2}, {0, 1}, 50] (* Vincenzo Librandi, Nov 27 2015 *)

PROG

(PARI) a(n)=if(n<0, 0, imag(quadgen(-7)^n))

(Sage) [lucas_number1(n, 1, +2) for n in xrange(0, 46)] # Zerinvary Lajos, Apr 22 2009

(MAGMA) [0] cat [n le 2 select 1 else Self(n-1)-2*Self(n-2): n in [1..45]]; // Vincenzo Librandi, Nov 27 2015

(PARI) x='x+O('x^100); concat(0, Vec(x/(1-x+2*x^2))) \\ Altug Alkan, Dec 04 2015

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 December 3 12:43 EST 2016. Contains 278735 sequences.