oeis-0.3.4: Interface to the Online Encyclopedia of Integer Sequences (OEIS)

Math.OEIS

Description

Interface to the Online Encyclopedia of Integer Sequences (OEIS). See http://oeis.org/.

Synopsis

# Example usage

Suppose we are interested in answering the question, "how many distinct binary trees are there with exactly 20 nodes?" Some naive code to answer this question might be as follows:

``` import Data.List (genericLength)

-- data-less binary trees.
data BTree = Empty | Fork BTree BTree  deriving Show

-- A list of all the binary trees with exactly n nodes.
listTrees :: Int -> [BTree]
listTrees 0 = [Empty]
listTrees n = [Fork left right |
k <- [0..n-1],
left <- listTrees k,
right <- listTrees (n-1-k) ]

countTrees :: Int -> Integer
countTrees = genericLength . listTrees
```

The problem, of course, is that `countTrees` is horribly inefficient:

```*Main> :set +s
*Main> countTrees 5
42
(0.00 secs, 0 bytes)
*Main> countTrees 10
16796
(0.47 secs, 27513240 bytes)
*Main> countTrees 12
208012
(7.32 secs, 357487720 bytes)
*Main> countTrees 13
*** Exception: stack overflow
```

There's really no way we can evaluate `countTrees 20`. The solution? Cheat!

``` import Math.OEIS

-- countTrees works ok up to 10 nodes.
-- [1,2,5,14,42,132,429,1430,4862,16796]
smallTreeCounts = map countTrees [0..10]

-- now, extend the sequence via the OEIS!
treeCounts = extendSequence smallTreeCounts
```

Now we can answer the question:

``` *Main> treeCounts !! 20
6564120420
```

Sweet. Of course, to have any sort of confidence in our answer, more research is required! Let's see what combinatorial goodness we have stumbled across.

```*Main> description `fmap` lookupSequence smallTreeCounts
Just "Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers."
```

Catalan numbers, interesting. And a nice formula we could use to code up a real solution! Hmm, where can we read more about these so-called 'Catalan numbers'?

```*Main> (head . references) `fmap` lookupSequence smallTreeCounts
Just ["A. Bernini, F. Disanto, R. Pinzani and S. Rinaldi, Permutations defining convex permutominoes, preprint, 2007."]
Just ["N. J. A. Sloane, <a href=\"http://www.research.att.com/~njas/sequences/b000108.txt\">The first 200 Catalan numbers</a>"]
```

And so on. Reams of collected mathematical knowledge at your fingertips! You must promise only to use this power for Good.

# Lookup functions

Look up a sequence in the OEIS by its catalog number. Generally this would be its A-number, but M-numbers (from the /Encyclopedia of Integer Sequences) and N-numbers (from the Handbook of Integer Sequences/) can be used as well.

Note that the result is not in the `IO` monad, even though the implementation requires looking up information via the Internet. There are no side effects to speak of, and from a practical point of view the function is referentially transparent (OEIS A-numbers could change in theory, but it's extremely unlikely).

Examples:

``` Prelude Math.OEIS> getSequenceByID "A000040"    -- the prime numbers
Just [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47...

Prelude Math.OEIS> getSequenceByID "nosuch"     -- no such sequence!
Nothing
```

Look up a sequence by ID number, returning a data structure containing the entirety of the information the OEIS has on the sequence.

The standard disclaimer about not being in the `IO` monad applies.

Examples:

``` Prelude Math.OEIS> description `fmap` lookupSequenceByID "A000040"
Just "The prime numbers."

Prelude Math.OEIS> keywords `fmap` lookupSequenceByID "A000105"
Just [Nonn,Hard,Nice,Core]
```

Extend a sequence by using it as a lookup to the OEIS, taking the first sequence returned as a result, and using it to augment the original sequence.

Note that `xs` is guaranteed to be a prefix of `extendSequence xs`. If the matched OEIS sequence contains any elements prior to those matching `xs`, they will be dropped. In addition, if no matching sequences are found, `xs` will be returned unchanged.

The result is not in the `IO` monad even though the implementation requires looking up information via the Internet. There are no side effects, and practically speaking this function is referentially transparent (technically, results may change from time to time when the OEIS database is updated; this is slightly more likely than the results of `getSequenceByID` changing, but still unlikely enough to be essentially a non-issue. Again, purists may use `extendSequence_IO`).

Examples:

``` Prelude Math.OEIS> extendSequence [5,7,11,13,17]
[5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71...
```
``` Prelude Math.OEIS> extendSequence [2,4,8,16,32]
[2,4,8,16,32,64,128,256,512,1024,2048,4096,8192...
```
``` Prelude Math.OEIS> extendSequence [9,8,7,41,562]   -- nothing matches
[9,8,7,41,562]
```

Find a matching sequence in the OEIS database, returning a data structure containing the entirety of the information the OEIS has on the sequence.

The standard disclaimer about not being in the `IO` monad applies.

The same as `getSequenceByID`, but with a result in the `IO` monad.

The same as `lookupSequenceByID`, but in the `IO` monad.

extendSequence_IO :: [Integer] -> IO [Integer]Source

The same as `extendSequence`, but in the `IO` monad.

The same as `lookupSequence`, but in the `IO` monad.

Look up a sequence in the OEIS using its search function.

Interpret a string as a OEIS request, and return the results as Strings.

# Data structures

data Language Source

Programming language that some code to generate the sequence is written in. The only languages indicated natively by the OEIS database are Mathematica and Maple; any other languages will be listed (usually in parentheses) at the beginning of the actual code snippet.

Constructors

 Mathematica Maple Other

Instances

 Show Language

data Keyword Source

OEIS keywords. For more information on the meaning of each keyword, see http://oeis.org/eishelp2.html#RK.

Constructors

 Base Bref Changed Cofr Cons Core Dead Dumb Dupe Easy Eigen Fini Frac Full Hard More Mult New Nice Nonn Obsc Sign Tabf Tabl Uned Unkn Walk Word

Instances

 Eq Keyword Read Keyword Show Keyword

Data structure for storing an OEIS entry. For more information on the various components, see http://oeis.org/eishelp2.html.

Constructors

 OEIS FieldscatalogNums :: [String]Catalog number(s), e.g. A000040, N1425. (%I) sequenceData :: SequenceDataThe actual sequence data (or absolute values of the sequence data in the case of signed sequences). (%S,T,U) signedData :: SequenceDataSigned sequence data (empty for sequences with all positive entries). (%V,W,X) description :: StringShort description of the sequence. (%N) references :: [String]List of academic references. (%D) links :: [String]List of links to more information on the web. (%H) formulas :: [String]Formulas or equations involving the sequence. (%F) xrefs :: [String]Cross-references to other sequences. (%Y) author :: StringAuthor who input the sequence into the database. (%A) offset :: IntSubscript/index of the first term. (%O) firstGT1 :: IntIndex of the first term > 1. (%O) programs :: [(Language, String)]Code that can be used to generate the sequence. (%p,t,o) extensions :: [String]Corrections, extensions, or edits. (%E) examples :: [String]Examples. (%e) keywords :: [Keyword]Keywords. (%K) comments :: [String]Comments. (%C)

Instances

 Show OEISSequence