A001607 a(n) = -a(n-1) - 2*a(n-2).

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

Author

N. J. A. Sloane

Keywords

• sign
• easy

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).

Crossrefs

Apart from signs, same as A077020.

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

Cf. A172250.

Programs

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

Mathematica

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

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))

SageMath

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

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