[seqfan] Re: Newbie discovery

[Ian Hutchinson]

Sorry, I didn’t know there was an issue with attachments. Here’s a link to the graph I tried to include: https://i.imgur.com/pv09fZz.png

> On Jan 9, 2021, at 12:54 AM, Nick Shapiro <sfgeek at gmail.com> wrote:
> 
> Interesting, thank you for posting. (At least in my newbie opinion) Reminds
> me a bit of the Van Eck sequence, which is beautifly explained by Dr Slone
> on Numberphile https://www.youtube.com/watch?v=etMJxB-igrc
> 
> This mailing list doesn't seem to allow attachments. Is there somewhere
> online you can post the graph images? I'd love to see.
> 
> Thanks!
> 
> 
>> On Fri, Jan 8, 2021 at 7:26 PM Ian Hutchinson <ianrh125 at gmail.com> wrote:
>> 
>> Dear Seqfans,
>> 
>> I recently came up with a sequence that seemed to generate
>> emergent patterns, which I was fascinated by. I decided to search the OEIS
>> with the first ~10 terms to see if it was already known, and to my surprise
>> it returned zero results. This is either because I made a brand new
>> discovery or because I don't know much about existing sequences, so I
>> thought I'd get a more knowledgeable opinion on this.
>> 
>> I came up with this while looking for a pattern that would avoid repetition
>> using simple rules. I came up with a formula in the form of  a(n) =
>> f(a(n-1), y), where f(x,y) can be any function that receives two integers
>> and outputs a single value, and y represents the number of terms before
>> f(n-1) that are equal to f(n-1). This ensures that for any integer k, the
>> first occurrence of k will be followed by f(k,0), the second will be
>> followed by f(k,1), the third by f(k,2), and so on. After trying a version
>> of this with bitxor(x, y) as the main function, I found some intriguing
>> results.
>> 
>> The first 15 terms are 0, 0, 1, 1, 0, 2, 2, 3, 3, 2, 0, 3, 1, 3, 0, and
>> already start to reveal an interesting pattern. Terms 1-5 switch between 0
>> and 1 until the third time 0 shows up, which causes the next term to be 2.
>> Terms 6-10 switch between 2 and 3 in the exact same way, since there were
>> no previous occurrences of 2 or 3. Terms 11-15 display a new behavior,
>> moving all over the range [0, 3] in a less orderly manner, ending after the
>> 5th occurrence of 0. This pattern of repeating everything in a new range
>> and then creating a more complicated interaction in the combined range
>> continues at a larger and larger scale, at which point it starts to
>> generate some striking graphs:
>> [image: XOR sequence.png]
>> (This is a plot of the first 3,416 terms, stopping right after the 65th
>> zero)
>> 
>> I also noticed a few other interesting properties about the resulting
>> sequence:
>> - The graph of the first n terms will resemble the graph of the first n/4
>> terms at sufficiently large values (n>100).
>> - The behavior of bitxor such that bitxor(n, 0) = n for any integer n
>> causes an upward sloping line to appear in the top right quadrant of the
>> above graph, as the series regularly returns to zero and other small
>> numbers.
>> - If computed to infinite terms, this sequence may contain every possible
>> pair of positive integers exactly once. This is based on a few specific
>> observations that are a little too long to list out here, but I'd be happy
>> to elaborate on them.
>> 
>> Have any of you seen a sequence like this before? I'm curious to see if the
>> overall structure of it has been used anywhere else.
>> 
>> Thanks,
>> Ian
>> 
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>> 
> 
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