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

Dale Gerdemann dale.gerdemann at gmail.com
Fri Nov 7 17:00:17 CET 2014


I remade my video now following the suggestion of using (n,0) rather than
(0,n):

https://www.youtube.com/watch?v=_qeBPOmBwAs&feature=youtu.be

For each n, I plot the seed pairs for 1,2,3,...,Lucas(n). The resulting
graphs look very much like some kind of "Proof Without Words," though I am
unsure how to interpret the geometric shapes.

The colors, as before, indicate relative frequency of occurrence. To the
color key in the upper right, I added for each frequency, the number of
seed pairs that occur with that frequency. So, for example, at the
14-second mark, one can see that the seed pairs for 1,2,3,..,29 consist of
4 pairs which occur once, three pairs that occur twice and one pair each
for frequencies 3,4,5 and 7. By examining these frequency of frequencies,
one can guess an identity of the form:

Lucas(n) = 1*Lucas(n-4) + 2*Lucas(n-5) + ... + (n-5)*Lucas(2) + (n-4) +
(n-3) + `(n-1)

This seems like an awkward identity to me, though perhaps there is a way to
beautify it.

Dale


On Fri, Nov 7, 2014 at 12:46 AM, Frank Adams-Watters <franktaw at netscape.net>
wrote:

> 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/
>>>>>
>>>>>
>>>>>
>>>>
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>>>
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>>>
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>>
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>>
>>
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