# [seqfan] Re: A very fast convergent alternating series for phi

Vladimir Shevelev shevelev at bgu.ac.il
Sun Feb 24 21:51:40 CET 2013

```Thank you very much, Ed and Jean-Paul!
I would like to add the following generalization.
Let N>=1 be integer. Consider sequence
a(0)=0,a(1)=1, for n>=2, a(n)=N*a(n-1)+a(n-2).
For example, for N=2 we have Pell numbers.
Then we have
(1)Expression a(n+1) via a(n): a(n+1) = (N*a(n) + sqrt((N^2+4)*a^2(n) + 4*(-1)^n))/2;
(2)a^2(n+1) - a(n)*a(n+2) = (-1)^n (Catalan-like formula);
(3)sum{k=1,...,n}(-1)^(k-1)/(a(k)*a(k+1)) = a(n)/a(n+1);
(4)sum{k>=1}(-1)^(k-1)/(a(k)*a(k+1)) = 1/phi_N, where phi_N=(N+sqrt(N^2+4))/2 is a "metallic" ratio (for N=1-golden ratio, for N=2-silver ratio, etc.);
(5)a(n)/a(n+1) = 1/phi_N + r(n), where |r(n)| < 1/(a(n+1)*a(n+2)).
Since, for N>=2, phi_N>sqrt(5), then the last approximation for N>1 is better, than 1/(sqrt(5)*n^2) with n:=a(n+1) and illustrates Hurwitz's theorem in case 1/phi_N, N >1.  Moreover, we see that, for an arbitrary large number L there exists an infinite set of quadratic irrational numbers on which the constant sqrt(5)  one can
change by L. Indeed, phi_N tends to infinity, as N goes to infinity, therefore,
phi_N>>sqrt(5).

Best regards,

----- Original Message -----
From: allouche at math.jussieu.fr
Date: Sunday, February 24, 2013 6:40
Subject: [seqfan] Re: A very fast convergent alternating series for phi
To: seqfan at list.seqfan.eu

> If I am not mistaken, Formula (13) in
> http://mathworld.wolfram.com/GoldenRatio.html
> does not seem to be attributed to Roselle,
> but Formula (12) is.
>
> http://mathworld.wolfram.com/FibonacciNumber.htmlas well as the
> equivalent Formula (92) which is attributed to Wells, or at
> least cited to be in his book]
>
> jpa
>
>
> "L. Edson Jeffery" <lejeffery2 at gmail.com> a écrit :
>
> > Very nice, Vladimir. Your formula seems to resemble equation
> (13) in
> > Wolfram ( http://mathworld.wolfram.com/GoldenRatio.html )
> which in turn
> > seems to be attributed there to B. Roselle, but apparently no
> reference to
> > literature is given.
> >
> > Ed Jeffery
> >
> > _______________________________________________
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