Sequence relating to Pell numbers; a new technique

[Thomas Baruchel]

On Sun, 3 Oct 2004 12:21:18 +0100 (BST), <creigh at o2online.de> wrote:
> The following is perhaps an interesting sequence relating to the Pell 
> numbers:
>
> (a(n)) = (2, 5, 13, 32, 78, 189, 457, 1104, 2666, 6437, 15541, 37520, )
>
> a(n) = A000129(n+2) + A000129(n+1) - A024537(n)
> or, using the existing formula,
> A024537(n+1) =  1/2 * (P(n+1)+P(n)+1), with P(n) = Pell numbers (A000129)
> we have
> a(n) = 1/2 * ( 2*P(n+2) + P(n+1) - P(n) - 1 )

Remember that P(n) can be computed with the help of the general formula
for computing the convergents. Thus,

P(n+2) = 2 P(n+1) + P(n)

and the formula above can be written:
a(n) = 1/2 * ( P(n+2) + 3 P(n+1) - 1)

or (by substituting again):
a(n) = 1/2 * ( 5 P(n+1) + P(n) - 1)

Regards,

-- 
Thomas Baruchel