[seqfan] Re: Code

[Matthijs Coster]

Hello,
I tried to find this nice result myself. It can be found easily by 
solving 6 equations with 6 variables:
Let Fn be the n-th Fibonacci number
We would like to find
F(n+2) =  a * Fn^5 + b * Fn^4 * F(n+1) + c * Fn^3 * F(n+1)^2 + d * Fn^2 
* F(n+1)^3 + e * Fn * F(n+1)^4 + f * F(n+1)^5
For n = 0,1,2,3,4,5 we get the matrix:
[ 0 0 0 0 0 1] [ 1 1 1 1 1 1] [ 1 2 4 8 16 32] [ 32 48 72 108 162 243] [ 
243 405 675 1125 1875 3125] [ 3125 5000 8000 12800 20480 32768]
the vector [a,b,c,d,e,f] can be found easily (and uniquely by
(1, 3, 1, -3, -1, 1)

Therefore Hans's result:

Fn =  1 * Fn^5 + 3 * Fn^4 * F(n+1) + 1 * Fn^3 * F(n+1)^2 +
-3 * Fn^2 * F(n+1)^3 + -1 * Fn * F(n+1)^4 + 1 * F(n+1)^5 - F(n+1)


comparable results can be found when the size of the matrix will be changed.

Matthijs



On 22-06-17 05:37, Sean A. Irvine wrote:
> Hi Peter, I don't know much about Mathematica, but it is usually 
> fairly easy to find online specifications of various bits for the 
> language. So for example see, 
> http://reference.wolfram.com/language/ref/Table.html.en In this case, 
> it is constructing a table of 10 elements, where each element is 
> computed by substitution a value of n into the expression. Having said 
> this, I do wish that a lot more of the programs in the OEIS were 
> clearer about what they were doing. I like the ones which are clear 
> enough that they can be easily translated into any other programming 
> language. Sean. On 22 June 2017 at 15:16, Peter Lawrence 
> <peterl95124 at sbcglobal.net> wrote:
>> Hans, As I am not familiar with Mathematica (I’m just a C/C++ 
>> programmer), would it be too much to ask for an explanation of this 
>> formula, in other words not a proof of its correctness, but rather 
>> the details of how it computes, what algorithm is being specified 
>> here ? Are F(n) and F(n+1) inputs, and if so then what is the output, 
>> F(n+2) ? Thanks, Peter Lawrence.
>>> On Jun 21, 2017, at 3:51 PM, seqfan-request at list.seqfan.eu wrote: 
>>> Message: 17 Date: Wed, 21 Jun 2017 15:32:01 -0400 From: Hans 
>>> Havermann <gladhobo at bell.net <mailto:gladhobo at bell.net>> To: 
>>> Sequence Fanatics Discussion list <seqfan at list.seqfan.eu <mailto: 
>> seqfan at list.seqfan.eu>>
>>> Subject: [seqfan] Code Message-ID: 
>>> <C6CC6D76-FE75-4251-ACCC-61CCD0F6C652 at bell.net <mailto: 
>> C6CC6D76-FE75-4251-ACCC-61CCD0F6C652 at bell.net>>
>>> Content-Type: text/plain; charset=us-ascii I presume that the 
>>> inclusion of code in sequences is to be able to 
>> generate/verify/extend that sequence. Simpler and faster is better. 
>> Browsing the OEIS today I chanced upon this Mathematica code for 
>> Fibonacci numbers < https://oeis.org/A000045 
>> <https://oeis.org/A000045> >:
>>> Table[Fibonacci[n]^5 - Fibonacci[1 + n] + 3 Fibonacci[n]^4 
>>> Fibonacci[1 + 
>> n] + Fibonacci[n]^3 Fibonacci[1 + n]^2 - 3 Fibonacci[n]^2 Fibonacci[1 
>> + n]^3 - Fibonacci[n] Fibonacci[1 + n]^4 + Fibonacci[1 + n]^5, {n, 1, 
>> 10}]
>>> It's an interesting Fibonacci identity worthy perhaps of mention in the 
>> comments (if it isn't already there) but I'm not sure it ought to be 
>> where it is. -- Seqfan Mailing list - http://list.seqfan.eu/ 
> -- Seqfan Mailing list - http://list.seqfan.eu/