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

[Vladimir Shevelev]

Thank you, Richard, for interesting references and Alexander for an important question!
 
>Could you generalize to
>a(n)=N*a(n-1)+M*a(n-2).
>?

>Regards,
>ARP

 
Here is an answer on Alexander's question.
 
For positive integers M,N, consider sequence  a(0)=0,a(1)=1, for n>=2, a(n)=N*a(n-1)+M*a(n-2). Then
(1) Expression a(n+1) via a(n): a(n+1) = (N*a(n) + sqrt((N^2+4*M)*a^2(n) + 4*(-1)^n)*M^n))/2;
(2) a^2(n+1) - a(n)*a(n+2) = (-M)^n (Catalan-like formula);
(3) sum{k=1,...,n}(-M)^(k-1)/(a(k)*a(k+1)) = a(n)/a(n+1);
(4) sum{k>=1}(-M)^(k-1)/(a(k)*a(k+1)) = 1/phi_(N, M), where phi_(N,M)=(N+sqrt(N^2+4*M))/2;
(5)  a(n)/a(n+1) = 1/phi_(N,M) + r(n), where |r(n)| < M^n/(a(n+1)*a(n+2)).
Thus r(n)=O((4*M/(N+sqrt(N^2+4*M))^2))^n).
However, it is easy to verify that function f_N(x)=(4*x/(N+sqrt(N^2+4*x))^2 is increasing for x>=1.  Therefore, the smallest r(n) we obtain in the case M=1.
 
Best regards,
Vladimir



----- Original Message -----
From: mathar <mathar at mpia-hd.mpg.de>
Date: Sunday, February 24, 2013 21:55
Subject: [seqfan] Re: A very fast convergent alternating series for phi
To: seqfan at seqfan.eu

> In response to http://list.seqfan.eu/pipermail/seqfan/2013-
> February/010846.html
> vs> 
> vs> (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.);  
> vs> ...
> 
> one should point to the applicable literature on the subject of
> summing inverse products of terms from the 2-term recurrences:
> 
> Gert Almkvist, A solution to a tantalizing problem, Fib. Quart. 
> (1986) 316, 
> http://www.fq.math.ca/Scanned/29-3/andre-jeannin1.pdf
> 
> Blagoj S. Popov, Summation of reciprocal series of numerical 
> functions of second order, Fib. Quart. 24 (1) (1986) 17-21, 
> http://www.fq.math.ca/Scanned/24-1/popov.pdf
> 
> R. Andre-Jeannin, Summation of certain reciprocal series related 
> to Fibonacci and Lucas numbers, Fig. Quart. 29 (1991) 200, 
> http://www.fq.math.ca/Scanned/29-3/andre-jeannin1.pdf
> 
> R. S. Melham, A generalization of a result of Andre-Jeannin 
> concering summation of reciprocals, Proguliae Mathematica 57 (1) 
> (2000) p. 45
> http://www.emis.ams.org/journals/PM/57f1/pm57f104.pdf
> 
> F. Zhao, Notes on reciprocal series related to Fibonacci and 
> Lucas Numbers, Fib. Quart. 39 (5) (2001) 392 , 
> http://www.fq.math.ca/Scanned/39-5/zhao1.pdf
> 
> N. Omur, On reciprocal series of generalized Fibonacci numbers 
> with subscripts in arithmetic progression, Discr. Dyn. Nat. Soc. 
> 2012 #684280 http://dx.doi.org/10.1155/2012/684280
> 
> RJM
> 
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> 

 Shevelev Vladimir‎