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A001607 a(n) = -a(n-1) - 2*a(n-2).
(Formerly M2225 N0883)
16
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; 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.

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, Oct 29 2003

The sequence a(n) is connected with the sequence A110512 (see Witula's comments to this one). - Roman Witula, Jul 27 2012

REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..500

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.

Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 82.

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

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

M. Mignotte, Propriétés arithmétiques des suites récurrentes, Besançon, 1988-1989, see p. 14. In French.

Eric Weisstein's World of Mathematics, Lehmer Number

Index entries for linear recurrences with constant coefficients, signature (-1,-2).

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, Feb 05 2003

a(n) = (1/7*I)*sqrt(7)*((-1/2-(1/2*i)*sqrt(7))^n - (-1/2+(1/2*i)*sqrt(7))^n), where i=sqrt(-1). - Paolo P. Lava, Jul 19 2011

x/(x^2+x+2) = Sum_{n>=0} a(n)*(x/2)^n. - Benoit Cloitre, Mar 12 2002

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

a(n+1) = Sum_{k=0..n} A172250(n,k)*(-1)^k. - Philippe Deléham, Feb 15 2012

G.f.: x - 2*x^2 + 2*x^2/(G(0)+1) where G(k) = 1 + x/(1 - x/(x - 1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 16 2012

MATHEMATICA

LinearRecurrence[{-1, -2}, {0, 1}, 60] (* Harvey P. Dale, Aug 21 2011 *)

PROG

(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))

(MAGMA) [ n eq 1 select 0 else n eq 2 select 1 else -Self(n-1)-2*Self(n-2): n in [1..50] ]; // Vincenzo Librandi, Aug 22 2011

CROSSREFS

Apart from signs, same as A077020.

Sequence in context: A188509 A265707 A188146 * A167433 A077020 A107920

Adjacent sequences:  A001604 A001605 A001606 * A001608 A001609 A001610

KEYWORD

sign,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 27 22:57 EDT 2017. Contains 287210 sequences.