Is it possible to invoke the OEIS from Mathematica?

Question

I had always wondered if there might be a way to write a function, which I'll call OEISData[], that more or less works as a curated data function for The On-Line Encyclopedia of Integer Sequences.

I would imagine that the usage might be a little something like this:

OEISData["A004001"][9]
5

OEISData["A003418"][Range[8, 15]]
840, 2520, 2520, 27720, 27720, 360360, 360360, 360360

OEISData["A005849", "Keywords"]
{"hard", "nonn", "nice", "more"}

An API or something to retrieve data from the OEIS site might be needed for an implementation of this function. Is a function like this possible, with what Mathematica is currently capable of?

Answer 1

There is a Mathematica package exactly for this at the OEIS wiki.

Somewhat related: there's also a package for formatting data into the OEIS format.

WolframAlpha also has some of this information, though I'm not sure how to get the $n^{\mathrm{th}}$ term of the sequence.

In[1] := WolframAlpha["A004001", {{"TermsPod:IntegerSequence", 1}, "ComputableData"}]

Out[1] = {1, 1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, 8, 8, 8, 8, 9, 10, 11,
          12, 12, 13, 14, 14, 15}

Or:

In[1] := WolframAlpha["A018900", {{"Continuation", 1}, "ComputableData"}]

Out[1] = {3, 5, 6, 9, 10, 12, 17, 18, 20, 24, 33, 34, 36, 40, 48, 65, 66, 68, 72}

Answer 2

A bit of a hack, could do with some polishing, but the basic idea will work:

OEISData[str_] := 
  StringSplit[#, ","] & /@ 
  Select[StringSplit[Import["http://oeis.org/search?q=" <> str]], 
  StringMatchQ[#, __ ~~ ","] &];

OEISData["A004001"][[9]]

If you just want the numbers, it could be even easier to just import from http://oeis.org/A004001/list (assuming that the input is a valid sequence identifier):

OEISSequence[str_] := ToExpression /@ 
        First@StringCases[Import["http://oeis.org/" <> str <> "/list"],
        "[" ~~ x__ ~~ "]" :> StringSplit[x, ","]];

Take[OEISSequence["A004001"], 20]

{1, 1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, 8, 8, 8, 8, 9, 10, 11, 12}

Answer 3

I liked Szabolcs’ answer but would like to remind about free form input here. We get so much information using it for very little typing. Plus we get native to M. format. For those who does not know this yet - at the beginning of new input line press equal sign “=” twice to get orange spiky and then type in free form. In this case you see result below. This is NOT web browser but M. notebook. Of course you can get the same on W|A website. But additionally here you can get the data. For example go to “Sequence terms” pod and click “more” to get a few more terms. Then press little plus sign in the top right corner and then and from the menu choose “computable data”. This pastes in M. notebook what you see here at the lower part of the image the image. And this also partially answers Szabolcs’ question about more terms ;-) This is also a good way to learn tricks of WolframAlpha[] function.

Answer 4

Here is the GitHub repo for my OEIS Mathematica package:

https://github.com/Psychedelic-Geometry/OEIS-Mathematica.git

Answer 5

I now have a ServiceConnection for this, to make things a little bit more standardized.

Installation

You can install it like so:

PacletInstall[
 "ServiceConnection_OEIS",
 "Site" ->
  "http://www.wolframcloud.com/objects/b3m2a1.paclets/PacletServer"
 ]

Then connect like:

$so = ServiceConnect["OEIS"]

Search

fibData = $so["Search", "Query" -> Fibonacci /@ Range[25]];
fibData[All, "Description"] // Normal

<|
     "000045" -> "Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.", 
     "212804" -> "Expansion of (1-x)/(1-x-x^2).", 
     "147316" -> "Fibonacci numbers (A000045) starting at offset -20.",
     "039834" -> "a(n+2) = -a(n+1) + a(n) (signed Fibonacci numbers) with a(-2) = a(-1) = 1; or Fibonacci numbers (A000045) extended to negative indices.", 
     "152163" -> "a(n)=a(n-1)+a(n-2), n>1 ; a(0)=1, a(1)=-1 .", 
     "236191" -> "(-1)^floor( (n-1) / 3 ) * F(n), where F = Fibonacci." 
    |>

$so["Search", "Query" -> {4, 12, 17, 25, 33, 38, 46}][All, "Description"] // Normal

<|
     "134860" -> "Wythoff AAB numbers; also, Fib101 numbers: those n for which the Zeckendorf expansion A014417(n) ends with 1,0,1.", 
     "095099" -> "Duplicate of A134860." 
    |>

SequenceData

fib2 = $so["SequenceData", "ID" -> "000045"];
fib2["Algorithms", "Mathematica"] // Normal

{
     "Table[Fibonacci[k], {k, 0, 50}] (* Mohammad K. Azarian, Jul 11 2015 *)", 
     "Table[2^n Sqrt @ Product[(Cos[Pi k/(n + 1)]^2 + 1/4), {k, n}] // FullSimplify, {n, 15}]; (* Kasteleyn's formula specialized, Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009 *)", 
     "LinearRecurrence[{1, 1}, {0, 1}, 40] (* Harvey P. Dale, Aug 03 2014 *)", 
     "Fibonacci[Range[0, 20]] (* Eric W. Weisstein, Sep 22 2017 *)", 
     "CoefficientList[Series[-(x/(-1 + x + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 22 2017 *)" 
    }

Sequence

$so["Sequence", "ID" -> "000045"] // Take[#, 100] & // ListPlot

Answer 6

It sounds like this might be helpful. The following notebook allows you to specify a sequence and automatically import a detailed list of matching entries from the OEIS:

http://www.brotherstechnology.com/math/oeis_mathematica.html

Answer 7

Instead of relying on WolframAlpha data, you can query the database directly, preferable through the /internal data format, documented here:

https://oeis.org/eishelp1.html

One such page is:

https://oeis.org/A000139/internal

For example, %t starts a line of mathematica code. We might do the following to query all of these lines from an entry:

OEISMathematica[a_String] := Module[{s},
   s = StringSplit[Import["https://oeis.org/" <> a <> "/internal"], 
     "\n"];
   StringReplace[
    Select[s, StringMatchQ[#, RegularExpression@"^%t.*$"] &], 
    RegularExpression@"^%t" -> ""]
   ];

(*this might take a while*)
ds = Dataset[
  AssociationMap[
   OEISMathematica, {"A000139", "A000142", "A004001", "A000142", 
    "A000165", "A000165", "A001044", "A001563", "A003422", "A009445", 
    "A010050", "A012245", "A033312", "A034886", "A038507", "A047920", 
    "A048631"}
   ]
  ]

(*try it!*)
StringJoin @@ ds["A048631"] // ToExpression

This is much better populated than the W|A data, but not curated to follow any standards either: Sometimes all of the lines might belong to the same example, sometimes they might be split across multiple lines (a sample often ends in a comment (**) though).

Answer 8

Here's my GitHub repo for an updated, standalone Mathematica notebook to query the OEIS. It allows the user to specify as many sequence terms as they'd like and return a table of matches with respective formulas and links to the corresponding OEIS pages:

https://github.com/HarlanBrothers/mathematica/find/main

Answer 9

Have a look at the output of

a = Entity["IntegerSequence", "A004001"];
Grid[List@@@Normal@DeleteMissing@AssociationMap[a, EntityProperties@a]]

Some of them have EntityProperty["IntegerSequence", "WolframLanguageTable"]. This seems to be a mathematica expression in String form which I assume to have the parameter n. It can be used as follows in a few cases I tested:

OEISDataTableFunction[a_String] := 
  Module[{t = 
     Entity["IntegerSequence", a]@
      EntityProperty["IntegerSequence", "WolframLanguageTable"]},
   If[MissingQ@t, Return@t];
   ToExpression["Function[{n}," <> t <> "]"]
   ];

n = 5;
"works, but n is not consistently the 'amount of terms'"
OEISDataTableFunction["A004001"][n]
OEISDataTableFunction["A000142"][n]
OEISDataTableFunction["A000165"][n]

"doesn't work:"
OEISDataTableFunction["A000140"][n]
OEISDataTableFunction["A000139"][n]

As you can see, the data is not particularly well curated.