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

Dale Gerdemann dale.gerdemann at gmail.com
Thu Nov 6 19:21:29 CET 2014

```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/
>>>>
>>>>
>>>
>>>
>> _______________________________________________
>>
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>>
>>
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```