| Safe Haskell | None |
|---|
Math.OEIS
Description
A Haskell interface to the Online Encyclopedia of Integer Sequences (OEIS), http://oeis.org/.
- 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)
- searchSequence_IO :: String -> IO (Maybe OEISSequence)
- lookupOEIS :: String -> IO [String]
- type SequenceData = [Integer]
- data Language
- = Mathematica
- | Maple
- | Other
- data Keyword
- 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. -- [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
getSequenceByID :: String -> Maybe SequenceData
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 "nosuch" -- no such sequence! Nothing
lookupSequenceByID :: String -> Maybe OEISSequence
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]
extendSequence :: SequenceData -> SequenceData
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
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.
getSequenceByID_IO :: String -> IO (Maybe SequenceData)
The same as getSequenceByID, but with a result in the IO monad.
lookupSequenceByID_IO :: String -> IO (Maybe OEISSequence)
The same as lookupSequenceByID, but in the IO monad.
extendSequence_IO :: [Integer] -> IO [Integer]
The same as extendSequence, but in the IO monad.
lookupSequence_IO :: SequenceData -> IO (Maybe OEISSequence)
The same as lookupSequence, but in the IO monad.
searchSequence_IO :: String -> IO (Maybe OEISSequence)
Look up a sequence in the OEIS using its search function
lookupOEIS :: String -> IO [String]
Interpret a string as a OEIS request, and return the results as Strings
Data structures
type SequenceData = [Integer]
data Language
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 |
data Keyword
OEIS keywords. For more information on the meaning of each keyword, see http://oeis.org/eishelp2.html#RK.
data OEISSequence
Data structure for storing an OEIS entry. For more information on the various components, see http://oeis.org/eishelp2.html.
Constructors
| OEIS | |
Fields
| |
Instances