A169998 a(0)=1, a(1)=1; thereafter a(n) = -a(n-1) - 2*a(n-2).

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, -5025499, -2603047

Offset

0,3

Comments

Cassels, following Nagell, shows that a(n) = +- 1 only for n = 1, 2, 3, 5, 13.

The sequences A001607, A077020, A107920, A167433, and this one are all essentially the same except for signs.

References

J. W. S. Cassels, Local Fields, Cambridge, 1986, see p. 67.

Links

Table of n, a(n) for n=0..45.

F. Beukers, The multiplicity of binary recurrences, Compositio Mathematica, Tome 40 (1980) no. 2 , p. 251-267. See Theorem 2 p. 259.

M. Mignotte, Propriétés arithmétiques des suites récurrentes, Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (1989), article no. 3, 29 p., see p. 14. In French.

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

Formula

G.f.: (1 + 2*x) / (1 + x + 2*x^2). - R. J. Mathar, Jul 14 2011

a(n) = (i/sqrt(7))*(((-1 + i*sqrt(7))/2)^(n+2) - ((-1 -i*sqrt(7))/2)^(n+2)), where i=sqrt(-1). - Taras Goy, Jan 20 2026

E.g.f.: exp(-x/2)*(sqrt(7)*cos(sqrt(7)*x/2) + 3*sin(sqrt(7)*x/2))/sqrt(7). - Stefano Spezia, Jan 20 2026

Maple

f:=proc(n) option remember; if n <= 1 then 1 else -f(n-1)-2*f(n-2); fi; end;

Mathematica

LinearRecurrence[{-1, -2}, {1, 1}, 46] (* Jean-François Alcover, Feb 23 2024 *)

Prog

(PARI) a(n)=([0, 1; -2, -1]^n*[1; 1])[1, 1] \\ Charles R Greathouse IV, Jun 11 2015

Crossrefs

Cf. A001607, A077020, A107920, A167433.

Sequence in context: A077020 A107920 A167433 * A326729 A171998 A343615

Adjacent sequences: A169995 A169996 A169997 * A169999 A170000 A170001

Keyword

sign,easy

Author

N. J. A. Sloane, Aug 29 2010

Status

approved