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

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

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

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.

Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 14.

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

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

a(n) = 2^((n-1)/2)*ChebyshevU(n-1, -1/(2*sqrt(2))). - G. C. Greubel, Mar 24 2024

a(n) = (i*(((-1 - i*sqrt(7))/2)^n - ((-1 + i*sqrt(7))/2)^n))/sqrt(7). - Alan Michael Gómez Calderón, Jul 09 2024; after T. D. Noe, Oct 29 2003

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
(SageMath)
A001607=BinaryRecurrenceSequence(-1, -2, 0, 1)
[A001607(n) for n in range(51)] # G. C. Greubel, Mar 24 2024

Crossrefs

Apart from signs, same as A077020.

Cf. A001607, A002249, A077021, A107920, A167433, A169998.

Cf. A172250.

Sequence in context: A360965 A336301 A188146 * A167433 A077020 A107920

Adjacent sequences: A001604 A001605 A001606 * A001608 A001609 A001610

Keyword

sign,easy

Author

N. J. A. Sloane

Status

approved