[seqfan] Re: Purely algorithmic number sequence identification

[Antti Karttunen]

On Tue, Feb 24, 2015 at 3:06 PM,  <seqfan-request at list.seqfan.eu> wrote:

> Message: 12
> Date: Mon, 23 Feb 2015 19:21:07 +0100
> From: Philipp Emanuel Weidmann <pew at worldwidemann.com>
> To: seqfan at list.seqfan.eu
> Subject: [seqfan]  Re: Purely algorithmic number sequence
>         identification
> Message-ID: <1424715667.2631.30.camel at worldwidemann.com>
> Content-Type: text/plain; charset="UTF-8"
>
> Well, it turns out the first eight elements of A000001 satisfy the, umm,
> "slightly exotic" recurrence relation
>
>   a(1) = 1
>   a(2) = 1
>   a(n) = Floor(a(n-2)*(2-Sin(2^n)))   for n >= 3
>
> ;)
>
> In earnest, while I doubt that brute forcing formulas will bring any
> insight into sequences that have baffled mathematicians for centuries
> with their irregularity, what might indeed be interesting is to run the
> system not on one sequence, but on tens of thousands, all of which have
> no closed-form expression associated with them (is there a way to query
> those on OEIS?). In a matter of days, Sequencer would likely return a
> hundred or so closed forms, some of which may prove correct, which could
> then be investigated rigorously.
>
> For such a search, I should probably also add a lot more combinatorial
> and number theoretic primitives to the formula generator – whenever I
> randomly browse around OEIS, most of the sequences seem to be counting
> problems of some kind.

Kudos for interesting development!

Yes, especially number theoretic primitives would be welcome, because
for now your system does not exactly shine on any such sequences. For
example, although it correctly identifies factorials:

http://www.sequenceboss.org/?q=1%2C2%2C6%2C24%2C120%2C720%2C5040

then for primorials https://oeis.org/A002110 it remains baffled:
http://www.sequenceboss.org/?q=1%2C+2%2C+6%2C+30%2C+210%2C+2310%2C+30030%2C+510510%2C+9699690

Neither any success with the squares of primes:
http://www.sequenceboss.org/?q=4%2C+9%2C+25%2C+49%2C+121%2C+169%2C+289

(Not to speak about any of A000005, A000010 or A000203).

Now, when testing the third row of "Ludic array"
https://oeis.org/A255127

http://www.sequenceboss.org/?q=5%2C+19%2C+35%2C+49%2C+65%2C+79%2C+95%2C+109%2C+125%2C+139%2C+155%2C+169

the SequenceBoss, guesses a working recurrence for it:

a_1 = 5, a_2 = 19, a3 = 35, a_n = a_{n-2} - a_{n-3} + a_{n-1} for n >= 4

(This mirrors the recurrence a(n) = a(n-1) + a(n-2) - a(n-3), n>=4
given for A007310 by Roger Bagula)


No such success with the later rows of A255127 though:

http://www.sequenceboss.org/?q=7%2C+31%2C+59%2C+85%2C+113%2C+137%2C+163%2C+191%2C+217%2C+241

http://www.sequenceboss.org/?q=11%2C++55%2C+103%2C+151%2C+203%2C+251%2C++299%2C++343%2C++391%2C++443

(although when looking at their graphs, they all look awfully linear...)


Cheers,

Antti

>
> Best regards
> Philipp
>