[HN Gopher] "A Handbook of Integer Sequences" Fifty Years Later
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"A Handbook of Integer Sequences" Fifty Years Later
Author : zeebeecee
Score : 125 points
Date : 2023-01-11 12:37 UTC (10 hours ago)
(HTM) web link (arxiv.org)
(TXT) w3m dump (arxiv.org)
| cscheid wrote:
| This makes me so happy to read. I had the privilege of working on
| the same lab as Neil (and Dave Applegate, another notable person
| in OEIS). No exaggeration at all to call them geniuses, you hang
| out with them for 5 minutes and know they're cut from different
| cloth. Nicest folk, too.
| dahart wrote:
| > My fascination with these sequences began in 1964 when I was a
| graduate student at Cornell University in Ithaca, NY, studying
| neural networks. I had encountered a sequence of numbers,
| 1,8,78,944,13800,..., and I badly needed a formula for the n-th
| term, in order to determine the rate of growth of the terms (this
| would indicate how long the activity in this very simple neural
| network would persist). I will say more about this sequence in
| Section 2.1.
|
| It's really fascinating to bump into mentions of NNs from the 60s
| & 70s. They seems to be quite hot at the time. The paper on the
| Medial Axis Transform mentions neural networks too, in a way that
| makes it seem like it was the cool thing to do. By the time I was
| in college, NNs were very out of fashion.
|
| Here's the NN problem Neil was working on, and the first sequence
| in the database: https://oeis.org/A000435
| zitterbewegung wrote:
| Yea neural networks were actually invented in the 40s by Warren
| McCulloch and Walter Pitts at University of Illinois at
| Chicago. They had a few isolated results until GPUs and
| distributed computation really kicked them into high gear and
| that made the change in terms to "deep learning" and now GPT-3
| and other networks are hyperparamaterized neural networks with
| millions to billions of parameters .
| ISL wrote:
| I was part of a research group that extensively trained small
| neural networks for image-processing in 2001, the high-energy
| physics community had been using them for many years by that
| time.
|
| Furthermore, I believe that the PalmPilot's handwriting-
| recognition engine also had a neural-network component.
|
| Agreed that the usage has increased radically in the last
| twenty years, but even before the GPU-based revolution, it
| felt like neural networks were already broadly known and in
| use across the sciences and engineering. They were just
| slower :).
| visarga wrote:
| True, but scaling has its own problems. It was necessary to
| find better optimisers, activation functions, regularisers,
| weight sharing schemes, architectures and many other
| ingredients to make it work. And to prepare the large
| datasets, and invent the whole stack of frameworks, from CUDA
| to HuggingFace.
|
| We have had 250,000 ML papers written since 2012. That's a
| lower bound on the number of distinct experiments necessary
| to find the winning tickets of today. Inventing the step-
| activated neuron formula was less than 1% of the way here.
| jacquesm wrote:
| The OEIS lives here:
|
| https://oeis.org/
|
| Super useful resource.
| typical182 wrote:
| As I understand it, written in Go.
|
| There's a mildly humorous "How do you know" exchange where
| someone on HN quizzes the very person most likely to know:
|
| https://news.ycombinator.com/item?id=9920020
| jacquesm wrote:
| HN has had a couple of those.
| recov wrote:
| Mandatory HN lore
| https://news.ycombinator.com/item?id=35079
| jacquesm wrote:
| That was one of the ones I had in mind. What's really
| neat is that _everybody_ from that thread is still
| visiting HN and participating.
| yreg wrote:
| I always have a need to use this on puzzle hunts, but I don't
| think it ever helped.
| jacquesm wrote:
| Quick: 2, 8, 18, 32, ?? ?
| eesmith wrote:
| 11 distinct answers on OEIS, assuming the 2 is either the
| first value, or you omitted an initial 0.
|
| Which were you thinking of?
| jacquesm wrote:
| The 2 is the first value. And no spoilers.
| ta123456789 wrote:
| OEIS foundation:
|
| http://oeisf.org/
|
| It's nice to see they have a solid plans on how to keep the
| website running indefinitely.
| dahart wrote:
| Yes, I'm also glad to hear it's future path is already paved.
| Sloane described the process and history in the paper:
|
| "In 2009, in order to ensure the long-term future of the
| database, I set up a non-profit foundation, The OEIS
| Foundation Inc., a 501(c)(3) Public Charity, whose purpose is
| to own, maintain and raise funds to support The On-Line
| Encyclopedia of Integer Se- quences or OEIS.
|
| On October 26, 2009, I transferred the intellectual property
| of The On-Line Ency- clopedia of Integer Sequences to the
| Foundation. A new OEIS with multiple editors was launched on
| November 11, 2010.
|
| Since then it has been possible for anyone in the world to
| propose a new sequence or an update to an existing sequence.
| To do this, users must first register, and then submissions
| are reviewed by the editors before they become a permanent
| part of the OEIS. Technically the OEIS is now a "moderated
| wiki".
|
| I started writing this article on November 11, 2022, noting
| that this marked twelve years of successful operation of the
| online OEIS, and also that the database is in its 59th year
| of existence."
| Aardwolf wrote:
| The one thing I wish is they had a keyword for base-ten
| related sequences (rather than only "base" for any base),
| because base ten related sequences simply are almost always
| going to be way more recreational maths related than base two
| or base three related sequences.
| totetsu wrote:
| Any website that lets me see "The numbers of Mozart's piano
| concerti" as a graph must be doing something correctly.
| http://oeis.org/A064172/graph
| Isamu wrote:
| A classic resource. I have my own favorite sequences. Thanks Neil
| for this unexpected way of connecting to previous research!
| Someone wrote:
| For real numbers, there's the dictionary of real numbers
| (https://www.amazon.com/Dictionary-Real-Numbers-Jonathan-
| Borw...), _"a list of just over 100,000 eight-digit real numbers
| in the interval [0,1) that arise as the first eight digits of
| special values of familiar functions"_
|
| Its online equivalent is the inverse symbolic calculator
| (https://en.wikipedia.org/wiki/Inverse_Symbolic_Calculator)
| lifthrasiir wrote:
| Or use ries: https://mrob.com/pub/ries/index.html
| anthk wrote:
| The series of dividing an integer over 7 are nice.
| ufo wrote:
| > It was no mind-reading trick, the Catalan numbers are certainly
| the most common sequence that people don't know about
|
| Guilty as charged! I learned about this sequence after looking it
| up in the OEIS, back when I was still a young student.
| dleather wrote:
| Is there something similar for real sequences?
| NeilSloane wrote:
| There's a version with fewer errors and typos here:
| http://neilsloane.com/doc/HIS50.pdf
| DonHopkins wrote:
| My favorite hard core nerd insult used to be "Your idea of a hot
| date is looking up dirty words in the unabridged dictionary," but
| now I'm going to use "Your idea of a hot date is looking up 69 in
| the Handbook of Integer Sequences."
| andreareina wrote:
| Neil Sloane (author of the paper and curator of the OEIS) has
| been featured on Numberphile several times and it's always a
| pleasure to watch.
| https://m.youtube.com/playlist?list=PLt5AfwLFPxWJXQqPe_llzWm...
| jl6 wrote:
| Seconded. He has an otherworldly curiosity.
| optimalsolver wrote:
| I'll take this opportunity to point out my favorite integer
| sequence, Recaman's Sequence:
|
| https://www.youtube.com/watch?v=FGC5TdIiT9U
| peter_d_sherman wrote:
| >"My fascination with these sequences began in 1964 when I was a
| graduate student at Cornell University in Ithaca, NY, studying
| neural networks. I had encountered a sequence of numbers, 1, 8,
| 78, 944, 13800, . . ., and I badly needed a formula for the n-th
| term, in order to determine the rate of growth of the terms..."
|
| Related Mathologer video:
|
| Mathologer - "Why don't they teach Newton's calculus of 'What
| comes next?'"
|
| https://www.youtube.com/watch?v=4AuV93LOPcE
| anderskaseorg wrote:
| The finite difference method of that video is only useful for
| finding _polynomial_ sequences. Of course, any finite sequence
| can be extended to some polynomial, but in many cases (such as
| this one) that's not the result you're looking for.
| OscarCunningham wrote:
| https://johndonleyva.tripod.com/DifferenceTables.htm
|
| > Robert Jackson suggests that if you've completed a
| difference table and still don't understand the sequence, you
| should turn the paper through an angle of 60 degrees, say,
| and start again and perhaps repeat this several times to make
| a fan of difference tables.
| peter_d_sherman wrote:
| Specifically, in this case, _why_ isn 't it?
| eesmith wrote:
| Because this sequences isn't polynomial. It's
| https://oeis.org/A000435 , with the explicit formula
| a(n) = (n-1)! * Sum_{k=0..n-2} n^k/k!
|
| and the approximate form shows it's grows roughly as n^n:
| a(n) ~ sqrt(Pi/2)*n^(n-1/2)
|
| Here's my Python implementation: from math
| import factorial from fractions import Fraction as F
| def A000435(n): return int(factorial(n-1) *
| sum(F(n**k, factorial(k)) for k in range(0, n-1)))
|
| The video you linked to is on OEIS at
| https://oeis.org/A000127 and is a quartic:
| def A000127(n): return (n**4 - 6*n**3 + 23*n**2 -
| 18*n + 24)//24
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