[HN Gopher] The astonishing behavior of recursive sequences
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The astonishing behavior of recursive sequences
Author : pseudolus
Score : 84 points
Date : 2023-11-17 11:54 UTC (11 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| continuitylimit wrote:
| Interesting how the nth term index for n-Sonos non-integral is
| itself monotonic. Intuitively it seems it doesn't have to be.
| (Handwavy looking at the n-S() behavior as hitting a
| 'catastrophic' transition at the n-th, so question is why is this
| index increasing as n increases?)
|
| https://oeis.org/A030127
| empath-nirvana wrote:
| https://www.youtube.com/watch?v=p-HN_ICaCyM
|
| As always, there is a numberphile video.
| snake_plissken wrote:
| Can someone explain why it goes from strictly
| multiplication/division for Somos 1,2 and 3 but then at Somos 4
| the addition operation is added? It seems completely arbitrary.
| davesque wrote:
| Here's a definition from Wolfram math world that includes an
| addition operation for any valid value of _k_ :
| https://mathworld.wolfram.com/SomosSequence.html
|
| Are you talking about the behavior of the floor operator in
| determining the number of terms in the sum? For _k_ in {1,2,3},
| _floor(k /2)_ = 1. So there would only be one term and thus no
| addition.
| convivialdingo wrote:
| Somewhat related, I have found The On-Line Encyclopedia of
| Integer Sequences[1] to be an amazing resource when I come across
| interesting numeric sequences.
|
| [1] https://oeis.org/
| uticus wrote:
| +1, amazing resource.
| nullc wrote:
| I've solved many gnarly problems by numerically building a
| sequence then looking up a formula that generalizes it in OEIS.
| Jeff_Brown wrote:
| That's such a cool idea. Do you then prove it is the formula
| you need, or do you just assume that if it matches for 10
| terms it's probably the formula?
|
| A math professor I had in college, Dave Kelly, had a trick
| where he'd show a seemingly reasonably-behaved sequence,
| explain where it came from, then show that the rest of the
| sequence was absurd -- like, it would be exponential for a
| while and then zero, or increase arithmetically for a while
| and then become undefined, etc. I wish I had taped that.
| jnellis wrote:
| Rarely have I found it to be that easy. Generally you'll
| get a hit on a partial match of your sequence but from that
| you'll learn about some number theory regarding that
| sequence or there will be links, formulas and code snippets
| that help you further research your specific algorithm
| you're trying to solve. You're just looking for that 'aha!'
| moment and this database usually delivers. Sometimes that
| 'aha' is the realization that it's going to be impossible
| or take too much time or resources to compute and that's
| just as helpful.
|
| Recursive/fractal sequences are toughest because there's
| very likely no direct computation. Say your sequence
| matches with a fractal sequence at only every fifth
| element, you still have to compute the elements you don't
| need to get the ones you do need. Sometimes you need a
| number theory math library to do the computation and that
| kind of dependency is overkill so now you've got to figure
| out how recreate just the parts of that math library you
| need or discover a shortcut (again, using the sequence
| database.) Invariably you'll discover something new and be
| able to contribute back.
| uticus wrote:
| I used to think mathematics was about counting. Now I know
| better: it's about perspective.
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