1

Consider the sequence -1, 0, 2, 5, 7, 10, etc. where you add 2, then 3 and repeat. Somehow, it could be noted like "-1, 0, (+2, +3)*" à la regular expression. I can say this is an increasing integer sequence where the difference of two consecutive elements is ultimately periodic.

But clearly, there must be a shorter name to this kind of integer sequences somewhere in mathematic literature.

And even maybe there is already some short notation like I did with "-1, 0, (+2, +3)*".

Can you provide bibliographic references for such integer sequences, please ?

Laurent Lyaudet
  • 163
  • 3
    Many things lack simple names. Here, while there might well be something somewhere, there's nothing universally recognized (so far as I know) so you'd just have to define it anyway. – lulu Oct 30 '23 at 23:47
  • 1
    You could call it "eventually arithmetic". But there's no fancy name nor notation. – Anthony Oct 30 '23 at 23:51
  • 1
    I think your description ("differences are eventually periodic") is already very informative and succinct, which suggests that's a good name to give these sequences. You could relate them to the idea of the discrete derivative. A characterisation of these sequences is "there is some $k$ such that $(x_{nk + j})_n$ is eventually an arithmetic progression, for each $j = 1, \dotsc, k$". – Izaak van Dongen Oct 31 '23 at 00:16
  • 1
    I'm curious about where you have encountered such sequences (which I characterize as eventually a union of arithmetic progressions) since I see them in my current work. If you care to answer you can find my email in my SE profile. – Ethan Bolker Oct 31 '23 at 00:25
  • Thanks for your comments :). I hoped there would be a known short name like "ultimately additively periodic sequences" or something. Maybe it's hidden in a 19th century article that is not in English and not available freely on the Internet. At least I know if it exists that it is not a common name. – Laurent Lyaudet Oct 31 '23 at 20:48
  • @EthanBolker I think you will be very surprised. I made up this particular sequence as an exemple for this question. Now the reason I asked this question is more complicated. I am a fullstack dev doing business logic for transportation, logistics and e-commerce. But I have a Ph.D. in theoretical computer science / discrete mathematics (graph theory). So I over-generalize what I see at my work. The problem was to handle an extra-delay of a few seconds at each delivery waypoint when the number of bags is between two integers per tranche of a smaller integer. – Laurent Lyaudet Oct 31 '23 at 20:55
  • Like add an extra-delay of 20 seconds for each tranche of 3 bags from 0 bags to 10 bags, hence [0,3) bags + 20 s, [3,6) bags + 40 s, etc. And I asked myself if there was already a nice notation for this kind of periodic sequences of intervals. I thought it would be unheard of, and downgraded my question to look for integer sequences first. – Laurent Lyaudet Oct 31 '23 at 20:58
  • I came up with this notation for intervals (it may be reals intervals instead of integers intervals, or ordinal intervals, etc.) : I use the constants \i for included and \e for excluded. If you want [0,3), [3,6), [6,9), etc. 0(+0\i, +3\e)* may be a compact notation. – Laurent Lyaudet Oct 31 '23 at 21:02
  • If you want [0,3], [5,10), [13,16], [18,23), etc., you can express it with -3(+3\i,+3\i,+2\i,+5\e). I made up these examples. But I see new ideas everywhere and if I had the time I would made this an RFC :). Periodic intervals are very common. I encountered them many times in accounting and taxation when you compute the cost of something. For the example of the extra-delay for bags, I was wondering on short notations like 0(+0\i, +3\e:result+=20) or 0(+0\i, +3\e:result=20 x iter)*. – Laurent Lyaudet Oct 31 '23 at 21:09

1 Answers1

2

I'm not sure if this answers your question in the way you're looking for, but the sequence you have listed is described by the Online Encyclopedia of Integer Sequences (https://oeis.org) as "Integers that are congruent to $\{2, 3\}$ mod $5$". The specific page for your sequence is https://oeis.org/A047215 and I definitely recommend checking out the site more since it's very cool.

There are a number of formulas listed on the above article which might help you characterize your sequence in a broader sense, or just without mod arithmetic if that's what you're looking for.

Hope this helps!

haeloune
  • 21
  • 3
    Hi! Welcome to MSE. If I understand the OP correctly, then the sequence would continue $-1, 0, 2, 5, 7, 10, 12, 15, 17, 20, 22, 25, ...$, so it's not quite related to the triangular numbers. I think the idea is that eventually the $2n$th term is given by adding $3$ to the previous term, and the $(2n + 1)$th term is given by adding $2$ the previous term. I agree that the OEIS is very cool though :) – Izaak van Dongen Oct 31 '23 at 00:10
  • 2
    @IzaakvanDongen ahh thank you, I see I was a little too excited to answer a question. There is, however, an entry on OEIS for the sequence listed by the op so I'll update it with that! – haeloune Oct 31 '23 at 00:13
  • Thanks, I did check on the Internet in general and on the OEIS in particular before asking. But the sequence I gave was just an example I made out :). Still, your answer is nice for other readers that do not know OEIS. – Laurent Lyaudet Oct 31 '23 at 20:44