Math.OEIS
 Contents Example usage Lookup functions Data structures
Description
A Haskell interface to the Online Encyclopedia of Integer Sequences (OEIS), http://www.research.att.com/~njas/sequences/. Comments, suggestions, or bug reports should be sent to Brent Yorgey, byorgey at gmail dot com.
Synopsis
getSequenceByID :: String -> Maybe SequenceData
lookupSequenceByID :: String -> Maybe OEISSequence
extendSequence :: SequenceData -> SequenceData
lookupSequence :: SequenceData -> Maybe OEISSequence
getSequenceByID_IO :: String -> IO (Maybe SequenceData)
lookupSequenceByID_IO :: String -> IO (Maybe OEISSequence)
extendSequence_IO :: [Integer] -> IO [Integer]
lookupSequence_IO :: SequenceData -> IO (Maybe OEISSequence)
type SequenceData = [Integer]
data Language
 = Mathematica | Maple | Other
data Keyword
 = Base | Bref | 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
data OEISSequence = OEIS {
 catalogNums :: [String] sequenceData :: SequenceData signedData :: SequenceData description :: String references :: [String] links :: [String] formulas :: [String] xrefs :: [String] author :: String offset :: Int firstGT1 :: Int programs :: [(Language, String)] extensions :: [String] examples :: [String] keywords :: [Keyword] comments :: [String]
}
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.
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
 getSequenceByID :: String -> Maybe SequenceData Source

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). If you're a nitpicky purist, feel free to use the provided getSequenceByID_IO instead.

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 "A-1"        -- no such sequence!
Nothing
```
 lookupSequenceByID :: String -> Maybe OEISSequence Source

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

Examples:

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

Prelude Math.OEIS> keywords `fmap` lookupSequenceByID "A000105"
Just [Nonn,Hard,Nice,Core]
```
 extendSequence :: SequenceData -> SequenceData Source

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]
```
 lookupSequence :: SequenceData -> Maybe OEISSequence Source

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

 getSequenceByID_IO :: String -> IO (Maybe SequenceData) Source
The same as getSequenceByID, but with a result in the IO monad.
 lookupSequenceByID_IO :: String -> IO (Maybe OEISSequence) Source
The same as lookupSequenceByID, but in the IO monad.
 extendSequence_IO :: [Integer] -> IO [Integer] Source
The same as extendSequence, but in the IO monad.
 lookupSequence_IO :: SequenceData -> IO (Maybe OEISSequence) Source
The same as lookupSequence, but in the IO monad.
Data structures
 type SequenceData = [Integer] Source
 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://www.research.att.com/~njas/sequences/eishelp2.html#RK.
Constructors
 Base Bref 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 OEISSequence Source
Data structure for storing an OEIS entry. For more information on the various components, see http://www.research.att.com/~njas/sequences/eishelp2.html.
Constructors
OEIS
 catalogNums :: [String] Catalog number(s), e.g. A000040, N1425. (%I) sequenceData :: SequenceData The actual sequence data (or absolute values of the sequence data in the case of signed sequences). (%S,T,U) signedData :: SequenceData Signed sequence data (empty for sequences with all positive entries). (%V,W,X) description :: String Short 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 :: String Author who input the sequence into the database. (%A) offset :: Int Subscript/index of the first term. (%O) firstGT1 :: Int Index 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