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

Bob Selcoe rselcoe at entouchonline.net
Wed Nov 5 22:25:55 CET 2014


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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