login
A001607
a(n) = -a(n-1) - 2*a(n-2).
(Formerly M2225 N0883)
19
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
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
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. A172250.
Sequence in context: A360965 A336301 A188146 * A167433 A077020 A107920
KEYWORD
sign,easy
STATUS
approved