I was reading some interesting questions about the topic "Can we make a program that, given a particular sequence, produces the next terms", like this one, and I really like the detailed answer of this one. I understand that the answer is "That's impossible without more restrictions", and that given some restrictions (polynomials, rational function or boolean map) we know some good algorithms, as the second answer I linked explains.
Now, a natural question is how much can we solve, trying our best even if we can't always solve it, to answer the original, general question. What I usually do when facing a hard sequence is trying to see if it's in OEIS, and if it seems to be there, seeing if there is any formula or algorithm to produce it in there. You can download a small version of OEIS with the first terms of each sequence, and you can make queries to find formulas or maple algorithms for a particular sequence. My question is, do you think it's feasible to download a small version of OEIS that includes, with the first terms, a little algorithm to produce it?
The natural problem here is that I haven't seen any link to download the entire database of OEIS with all the details, which maybe deserves its own question. Even if we had this, you need to read the formulas/algorithms (that can be written in different languages, from what I've seen) and interpret them correctly. But I thought maybe someone here knows how to solve this, in any case thanks in advance.
You could, as you note, download the sequences and their A-numbers from the link mentioned here: https://oeis.org/wiki/Welcome#Compressed_Versions
After searching that and finding one sequence (or a small number of sequences) of interest, you could scrape the respective page(s) for formulas. There are specific fields for Maple and Mathematica, which may be helpful, and otherwise, an entry in the PROGRAM field should include identifying information when it is not one of the standard languages with its own field in the database. See: http://oeis.org/wiki/Style_Sheet
Unofficially, but with the interests of the OEIS in mind, I would not recommend trying to download or scrape the OEIS in its entirety. Whether it's one person, or a whole host of people, we would certainly recommend using the compressed version of the database to identify sequences of interest by A-number first, then pulling their entire entry by scraping the site or querying the OEIS using methods that you have already mentioned: Programmatic access to On-Line Encyclopedia of Integer Sequences
If this sounds laborious, perhaps an alternative is the Wolfram Cloud, which actives this through other means. For example, you can navigate to the cloud (you may have to register just to get access) at: https://www.wolframcloud.com/
Typing in something like FindSequenceFunction[{1, 2, 3, 5, 17, 305, 34865}] will give you a formula, if Wolfram/Mathematica can find one. The documentation for FindSequenceFunction can be found here: https://reference.wolfram.com/language/ref/FindSequenceFunction.html
Wolfram/Mathematica can also invoke the OEIS using packages like the one described here: https://mathematica.stackexchange.com/questions/40/is-it-possible-to-invoke-the-oeis-from-mathematica