gistfile1.txt

Recurrencia a(n+2) + 2*a(n+1) + 2*a(n) = 0, n >= 0, a(0) = 1, a(1) = 3

[GP] (recuperando las raíces)

? polroots(x^2+2*x+2)
%1 = [-1.000000000000000000000000000 - 1.000000000000000000000000000*I, -1.000000000000000000000000000 + 1.000000000000000000000000000*I]~

[Haskell] 

(construyendo la recurrencia directamente)

> let t = 1 : 3 : zipWith (\ a b -> negate $ 2*a + 2*b) t (tail t)
> take 10 t

[1,3,-8,10,-4,-12,32,-40,16,48]

[Resolviendo la recurrencia]

a(n) = sqrt(2)^n [c1 sin (3pi/4 n) + c2 cos (3pi/4 n)]

sqrt(2) es el módulo de las raices conjugadas y 3pi/4 es el ángulo de una.

Como a(0) = 1, y sin(0) = 0, cos(0) = 1, c2 = 1. Luego, 

a(1) = 3 = sqrt(2) [c1 sin (3pi/4) + c2 cos (3pi/4)] = sqrt(2) [c1 (sqrt(2)/2) + (-sqrt(2)/2)] = c1 - 1, y c1 = 4.

[Haskell]

> let theta = (3*pi/4)
> let f n = (2**(n/2)) * ((4 * (sin (theta*n)))+(cos (theta*n)))
> let s = map f [0..]
> take 10 $ map round s

[1,3,-8,10,-4,-12,32,-40,16,48]

y las sucesiones 's' y 't' coinciden.

> take 100 $ zipWith3 (\ a b c -> a + 2*b + 2*c) (drop 2 s) (drop 1 s) s
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,1,0,-1,-2,-3,-2,-4,4,30,44,45,-19]

Hasta a(86) se cumple la recurrencia en la sucesión resuelta por (f n), después aparece error por punto flotante en seno, coseno y raíz.