oeis-0.3.10: Interface to the Online Encyclopedia of Integer Sequences (OEIS)

Math.OEIS

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."]
*Main> (head . links) fmap lookupSequence smallTreeCounts
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.

Look up sequences in the OEIS using its search function (returns up to 10 results).

Similar to lookupSequence_IO, but return up to 10 results.

# 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
 Source # Instance detailsDefined in Math.OEIS.Types Methods Source # Instance detailsDefined in Math.OEIS.Types MethodsshowList :: [Language] -> ShowS #

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
 Source # Instance detailsDefined in Math.OEIS.Types Methods(==) :: Keyword -> Keyword -> Bool #(/=) :: Keyword -> Keyword -> Bool # Source # Instance detailsDefined in Math.OEIS.Types Methods Source # Instance detailsDefined in Math.OEIS.Types MethodsshowList :: [Keyword] -> ShowS #

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
 Source # Instance detailsDefined in Math.OEIS.Types MethodsshowList :: [OEISSequence] -> ShowS #