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Re: help working with functions (follow-up)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg60035] Re: help working with functions (follow-up)
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Mon, 29 Aug 2005 01:38:39 -0400 (EDT)
  • Organization: The University of Western Australia
  • References: <dero92$s7j$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <dero92$s7j$1 at smc.vnet.net>, Bob Hanlon <hanlonr at cox.net> 
wrote:

> Amplifying on my previous response
> 
> f1[i_,j_,n_]:= Module[{k=0},
>       Nest[
>         Append[#,#[[-2]]+#[[-1]]+i*j^(k++)]&, 
>         {0,1}, n-2]];
> 
> f1[0,4,10]
> 
> {0,1,1,2,3,5,8,13,21,34}
> 
> You can also solve for the closed form
> 
> expr=FullSimplify[a[k]/.
>       RSolve[{a[k]==a[k-1]+a[k-2]+i*j^(k-2),a[0]==0,a[1]==1}, a[k], k][[1]],
>     Element[k, Integers]]
> 
> (2^(-k - 1)*(2*Sqrt[5]*(-(1 - Sqrt[5])^k + (1 + Sqrt[5])^k)*((j - 1)*j - 1) + 
>     i*(5*2^(k + 1)*j^k + 2*Sqrt[5]*((1 - Sqrt[5])^k - (1 + Sqrt[5])^k)*j + 
>     (-5 + 
> Sqrt[5])*(1 + Sqrt[5])^k - 
>       (1 - Sqrt[5])^k*(5 + Sqrt[5]))))/(5*((j - 1)*j - 1))
> 
> Consequently,
> 
> f2[i_,j_,n_]:=Table[(2^(-k-1)*(2*Sqrt[5]*(-(1-Sqrt[5])^k+(1+
>               Sqrt[5])^k)*((j-1)*j-1)+i*(5*2^(k+1)*j^k+2*Sqrt[5]*((
>                         1-Sqrt[5])^k-(1+Sqrt[5])^k)*j+(-5+Sqrt[5])*(
>                           1+Sqrt[5])^
>                   k-(1-Sqrt[
>                           5])^k*(5+Sqrt[5]))))/(5*((j-1)*j-1)),{
>                             k,0,n-1}]//Simplify;
> 
> Checking that f1 and f2 are equivalent
> 
> And@@Flatten@Table[f1[i,j,n]==f2[i,j,n],{i,5},{j,5},{n,2,10}]
> 
> True
> 
> For i=0 the sequence is the Fibonacci sequence

You can use this observation to rewrite expr as

  (i (-((1/2)(1 + Sqrt[5]))^k + j^k))/(-1 - j + j^2) + 

     (1 + (i (1 + Sqrt[5] - 2j))/(2 (-1 - j + j^2))) Fibonacci[k]

which is somewhat simpler, and clearly equal to Fibonacci[k] when i = 0.

Cheers,
Paul

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul


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