# [seqfan] Re: Smallest index of Fibonacci-like sequence containing n

Frank Adams-Watters franktaw at netscape.net
Fri Nov 7 00:46:10 CET 2014

```First, I agree with Allan that it is better to take pairs as (n,0)
rather than (0,n). It makes the definition simpler and simplifies some
manipulations. For example, the rule for finding the pair can be stated
as "run the sequence backward until the first value is greater than the
second".

Note also that in the Zeckendorf or "base Fibonacci" representation,
every Fibonacci-type sequence of positive integers eventually becomes
"[xxx], [xxx]0, [xxx]00, ...." For example, starting with 3,1, we get
3,1,4,5,9,14,23,37,..., which in base Fibonaci is
100,1,101,1000,10001,100001,1000010,10000100,... And the second number
before that pattern starts (5 in this case) uniquely specify the
sequence, with each non-negative integer occurring exactly once. This
thus provides a correspondence between the seed pairs and the
non-negative integers.

You really need to spend some time studying the Wythoff array to
understand what is going on with stuff. It's really beautiful.

Franklin T. Adams-Watters

-----Original Message-----
From: Dale Gerdemann <dale.gerdemann at gmail.com>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Thu, Nov 6, 2014 5:15 pm
Subject: [seqfan] Re: Smallest index of Fibonacci-like sequence
containing n

Hello Seqfans,

It's also interesting to look at the sequence of seed pairs. To make the
seed pair unique, take the pair (a,b) with the smallest value for a:

(0, 0), (0, 1), (0, 1), (0, 1), (0, 2), (0, 1), (0, 2), (2, 1), (0, 1),
(0,
3), (0, 2), (2, 1), (0, 4), (0, 1), (3, 1), (0, 3), (0, 2), (4, 1), (2,
1),
(3, 2),...

In https://www.youtube.com/watch?v=XAzDlDXOmbQ, I plotted these pairs
and
color-coded the frequency of occurrence as the sequence is extended.

Dale

On Wed, Nov 5, 2014 at 10:25 PM, Bob Selcoe <rselcoe at entouchonline.net>
wrote:

> A very interesting sequence and pretty easy to compute by hand (up to
a
> point).  The seed values which do not produce totally redundant
numbers are
> [0,k] k>=1 and [j,k]  j>k>0.  That is:
>
> [01,], [0,2], [0,3], [2,1], [0,4], [3,1], [0,5], [3,2], [4,1]...
>
> Progressing along the sequence of seed pairs, once a non-repeated
number
> is reached, all subsequent numbers are of the smallest index, used as
a(n).
> That alone should generate some interesting patterns and
observations, I
> think.
>
> The sequence with all positive seeds has similar properties, except
seed
> pairs are [j,k]  j>=k>0.  That is:
>
> [1,1], [2,1], [2,2], [3,1], [3,2], [4,1], [3,3]...   and I think is
> equally interesting.
>
> I'll submit this sequence unless Allan wants to, since it's basically
his
> idea.
>
> Cheers,
> Bob Selcoe
>
> --------------------------------------------------
> From: "Charles Greathouse" <charles.greathouse at case.edu>
> Sent: Wednesday, November 05, 2014 2:48 PM
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Smallest index of Fibonacci-like sequence
containing
> n
>
>  My script
>>
>> a(n)=if(n<2,return(n));for(k=1,n-1,for(a=0,k-1,my(A=a,B=k-
>> A);while(B<n,[A,B]=[B,A+B]);if(B==n,return(k))));n
>>
>> agrees with your hand-computed terms, and verifies that the sequence
is
>> not
>> in the encyclopedia.
>>
>> Charles Greathouse
>> Analyst/Programmer
>> Case Western Reserve University
>>
>> On Wed, Nov 5, 2014 at 3:45 PM, Charles Greathouse <
>> charles.greathouse at case.edu> wrote:
>>
>>  I think this sequence is interesting. Some quick observations: 0 is
the
>>> only number with index 0, all positive integers have index at least
1.
>>> If a
>>> number is k times a Fibonacci number, then its index is at most k
via (A,
>>> B) = (0, k); in particular, since 1 is a Fibonacci number, a(n) <=
n.
>>> (This
>>> suggests A054495 as a cross-reference.)
>>>
>>> Charles Greathouse
>>> Analyst/Programmer
>>> Case Western Reserve University
>>>
>>> On Wed, Nov 5, 2014 at 1:50 PM, Allan Wechsler <acwacw at gmail.com>
wrote:
>>>
>>>  Any two non-negative integers can seed a Fibonacci-like sequence,
F[0] =
>>>> A,
>>>> F[1] = B, F[i+2] = F[i+1] + F[i].
>>>>
>>>> Let A+B be called the "index" of this sequence.
>>>>
>>>> Of all Fibonacci-like sequences containing, say, 18, the one with
the
>>>> smallest index is {2,1,3,4,7,11,18...}, with an index of 3. So I
say
>>>> A[18]
>>>> = 3.
>>>>
>>>> If n is a classic Fibonacci number, A[n] = 1. If n is a Lucas
number
>>>> (like
>>>> 18), then A[n] = 3. If n is twice a Fibonacci number (like 16) then
>>>> A[n] =
>>>> 2.
>>>>
>>>> I have calculated A[n] by hand for n from 1 to 24. It is quite
possible
>>>> that I have made mistakes, but the sequence I get is:
>>>>
>>>> {1,1,1,2,1,2,3,1,3,2,3,4,1,4,3,2,5,3,5,4,1,6,4,3, ...}
>>>>
>>>> This is not in OEIS. I would've submitted it, but I would like
somebody
>>>> else to check my arithmetic first, because it seems unlikely that
such a
>>>> simple concept wouldn't have been entered already. If nobody steps
up to
>>>> the place quickly I will cobble together some code and submit
anyway.
>>>> Thanks for your assistance, seqfans!
>>>>
>>>> _______________________________________________
>>>>
>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>
>>>>
>>>
>>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>>
> _______________________________________________
>
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