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

Author

N. J. A. Sloane, Aug 29 2010

Keywords

• sign
• easy

Comments

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

The sequences A001607, A077020, A107920, A167433, A169998 are all essentially the same except for signs.

References

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

Links

• 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, Besançon, 1988-1989, see p. 14. In French.
• Index entries for linear recurrences with constant coefficients, signature (-1,-2).

Programs

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 *)

PARI

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

Formula

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