module Math.OEIS
  ( -- * Example usage
    -- $sample

    -- * Lookup functions
    getSequenceByID,    lookupSequenceByID
  , extendSequence,     lookupSequence
  , getSequenceByID_IO, lookupSequenceByID_IO
  , extendSequence_IO,  lookupSequence_IO
  , searchSequence_IO,  lookupOEIS
  , searchSequences_IO, lookupSequences_IO

    -- * Data structures
  , SequenceData
  , Language(..), Keyword(..)
  , OEISSequence(..)
  ) where

--------------------------------------------------------------------------------

import Control.Applicative ((<$>))
import Prelude
import Data.Char        (isDigit, isSpace)
import Data.List        (isPrefixOf, tails, find)
import Data.Maybe       (listToMaybe, fromMaybe)
import Network.URI      (escapeURIString, isAllowedInURI)
import System.IO.Unsafe (unsafePerformIO)

import Math.OEIS.Internal
import Math.OEIS.Types

--------------------------------------------------------------------------------

-- | Interpret a string as a OEIS request, and return the results as Strings.
lookupOEIS :: String -> IO [String]
lookupOEIS :: String -> IO [String]
lookupOEIS String
a = do
         let a' :: String
a'  = String -> String
commas (String -> String) -> (String -> String) -> String -> String
forall b c a. (b -> c) -> (a -> b) -> a -> c
. String -> String
forall a. [a] -> [a]
reverse (String -> String) -> (String -> String) -> String -> String
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (Char -> Bool) -> String -> String
forall a. (a -> Bool) -> [a] -> [a]
dropWhile Char -> Bool
isSpace (String -> String) -> (String -> String) -> String -> String
forall b c a. (b -> c) -> (a -> b) -> a -> c
. String -> String
forall a. [a] -> [a]
reverse (String -> String) -> (String -> String) -> String -> String
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (Char -> Bool) -> String -> String
forall a. (a -> Bool) -> [a] -> [a]
dropWhile Char -> Bool
isSpace (String -> String) -> String -> String
forall a b. (a -> b) -> a -> b
$ String
a
         Maybe OEISSequence
x <- String -> IO (Maybe OEISSequence)
searchSequence_IO String
a'
         case Maybe OEISSequence
x of
            Maybe OEISSequence
Nothing -> [String] -> IO [String]
forall a. a -> IO a
forall (m :: * -> *) a. Monad m => a -> m a
return [String
"Sequence not found."]
            Just OEISSequence
s  -> [String] -> IO [String]
forall a. a -> IO a
forall (m :: * -> *) a. Monad m => a -> m a
return [OEISSequence -> String
description OEISSequence
s, SequenceData -> String
forall a. Show a => a -> String
show (SequenceData -> String) -> SequenceData -> String
forall a b. (a -> b) -> a -> b
$ OEISSequence -> SequenceData
sequenceData OEISSequence
s]
 where commas :: String -> String
commas []                     = []
       commas (Char
x:Char
' ':String
xs) | Char -> Bool
isDigit Char
x = Char
x Char -> String -> String
forall a. a -> [a] -> [a]
: Char
',' Char -> String -> String
forall a. a -> [a] -> [a]
: String -> String
commas String
xs
       commas (Char
x:String
xs)                 = Char
x Char -> String -> String
forall a. a -> [a] -> [a]
: String -> String
commas String
xs

-- | Look up a sequence in the OEIS using its search function.
searchSequence_IO :: String -> IO (Maybe OEISSequence)
searchSequence_IO :: String -> IO (Maybe OEISSequence)
searchSequence_IO String
x = [OEISSequence] -> Maybe OEISSequence
forall a. [a] -> Maybe a
listToMaybe ([OEISSequence] -> Maybe OEISSequence)
-> IO [OEISSequence] -> IO (Maybe OEISSequence)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> String -> IO [OEISSequence]
searchSequences_IO String
x

-- | Look up sequences in the OEIS using its search function (returns up to
-- 10 results).
searchSequences_IO :: String -> IO [OEISSequence]
searchSequences_IO :: String -> IO [OEISSequence]
searchSequences_IO String
x = (String -> String) -> String -> IO [OEISSequence]
forall a. (a -> String) -> a -> IO [OEISSequence]
getOEIS (String
baseSearchURI String -> String -> String
forall a. [a] -> [a] -> [a]
++) ((Char -> Bool) -> String -> String
escapeURIString Char -> Bool
isAllowedInURI String
x)

-- | 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

getSequenceByID :: String -> Maybe SequenceData
getSequenceByID :: String -> Maybe SequenceData
getSequenceByID = IO (Maybe SequenceData) -> Maybe SequenceData
forall a. IO a -> a
unsafePerformIO (IO (Maybe SequenceData) -> Maybe SequenceData)
-> (String -> IO (Maybe SequenceData))
-> String
-> Maybe SequenceData
forall b c a. (b -> c) -> (a -> b) -> a -> c
. String -> IO (Maybe SequenceData)
getSequenceByID_IO

-- | The same as 'getSequenceByID', but with a result in the 'IO' monad.
getSequenceByID_IO :: String -> IO (Maybe SequenceData)
getSequenceByID_IO :: String -> IO (Maybe SequenceData)
getSequenceByID_IO String
x = (Maybe OEISSequence -> Maybe SequenceData)
-> IO (Maybe OEISSequence) -> IO (Maybe SequenceData)
forall a b. (a -> b) -> IO a -> IO b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap ((OEISSequence -> SequenceData)
-> Maybe OEISSequence -> Maybe SequenceData
forall a b. (a -> b) -> Maybe a -> Maybe b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap OEISSequence -> SequenceData
sequenceData) (String -> IO (Maybe OEISSequence)
lookupSequenceByID_IO String
x)

-- | 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]

lookupSequenceByID :: String -> Maybe OEISSequence
lookupSequenceByID :: String -> Maybe OEISSequence
lookupSequenceByID = IO (Maybe OEISSequence) -> Maybe OEISSequence
forall a. IO a -> a
unsafePerformIO (IO (Maybe OEISSequence) -> Maybe OEISSequence)
-> (String -> IO (Maybe OEISSequence))
-> String
-> Maybe OEISSequence
forall b c a. (b -> c) -> (a -> b) -> a -> c
. String -> IO (Maybe OEISSequence)
lookupSequenceByID_IO

-- | The same as 'lookupSequenceByID', but in the 'IO' monad.
lookupSequenceByID_IO :: String -> IO (Maybe OEISSequence)
lookupSequenceByID_IO :: String -> IO (Maybe OEISSequence)
lookupSequenceByID_IO String
x = [OEISSequence] -> Maybe OEISSequence
forall a. [a] -> Maybe a
listToMaybe ([OEISSequence] -> Maybe OEISSequence)
-> IO [OEISSequence] -> IO (Maybe OEISSequence)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> (String -> String) -> String -> IO [OEISSequence]
forall a. (a -> String) -> a -> IO [OEISSequence]
getOEIS String -> String
idSearchURI String
x

-- | 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]
extendSequence :: SequenceData -> SequenceData
extendSequence :: SequenceData -> SequenceData
extendSequence = IO SequenceData -> SequenceData
forall a. IO a -> a
unsafePerformIO (IO SequenceData -> SequenceData)
-> (SequenceData -> IO SequenceData)
-> SequenceData
-> SequenceData
forall b c a. (b -> c) -> (a -> b) -> a -> c
. SequenceData -> IO SequenceData
extendSequence_IO

-- | The same as 'extendSequence', but in the 'IO' monad.
extendSequence_IO :: [Integer] -> IO [Integer]
extendSequence_IO :: SequenceData -> IO SequenceData
extendSequence_IO [] = SequenceData -> IO SequenceData
forall a. a -> IO a
forall (m :: * -> *) a. Monad m => a -> m a
return []
extendSequence_IO SequenceData
xs = do
    Maybe OEISSequence
oeis <- SequenceData -> IO (Maybe OEISSequence)
lookupSequence_IO SequenceData
xs
    SequenceData -> IO SequenceData
forall a. a -> IO a
forall (m :: * -> *) a. Monad m => a -> m a
return (SequenceData -> IO SequenceData)
-> SequenceData -> IO SequenceData
forall a b. (a -> b) -> a -> b
$ case Maybe OEISSequence
oeis of
      Maybe OEISSequence
Nothing -> SequenceData
xs
      Just OEISSequence
s  -> SequenceData -> SequenceData -> SequenceData
extend SequenceData
xs (OEISSequence -> SequenceData
sequenceData OEISSequence
s)

-- | @extend xs ext@ returns the maximal suffix of @ext@ of which @xs@ is a
-- prefix, or @xs@ if @xs@ is not a prefix of any suffixes of @ext@. It is
-- guaranteed that
--
-- > forall xs ext. xs `isPrefixOf` (extend xs ext)
extend :: SequenceData -> SequenceData -> SequenceData
extend :: SequenceData -> SequenceData -> SequenceData
extend SequenceData
xs SequenceData
ext = SequenceData -> Maybe SequenceData -> SequenceData
forall a. a -> Maybe a -> a
fromMaybe SequenceData
xs (Maybe SequenceData -> SequenceData)
-> ([SequenceData] -> Maybe SequenceData)
-> [SequenceData]
-> SequenceData
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (SequenceData -> Bool) -> [SequenceData] -> Maybe SequenceData
forall (t :: * -> *) a. Foldable t => (a -> Bool) -> t a -> Maybe a
find (SequenceData
xs SequenceData -> SequenceData -> Bool
forall a. Eq a => [a] -> [a] -> Bool
`isPrefixOf`) ([SequenceData] -> SequenceData) -> [SequenceData] -> SequenceData
forall a b. (a -> b) -> a -> b
$ SequenceData -> [SequenceData]
forall a. [a] -> [[a]]
tails SequenceData
ext

-- | 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.
lookupSequence :: SequenceData -> Maybe OEISSequence
lookupSequence :: SequenceData -> Maybe OEISSequence
lookupSequence = IO (Maybe OEISSequence) -> Maybe OEISSequence
forall a. IO a -> a
unsafePerformIO (IO (Maybe OEISSequence) -> Maybe OEISSequence)
-> (SequenceData -> IO (Maybe OEISSequence))
-> SequenceData
-> Maybe OEISSequence
forall b c a. (b -> c) -> (a -> b) -> a -> c
. SequenceData -> IO (Maybe OEISSequence)
lookupSequence_IO

-- | The same as 'lookupSequence', but in the 'IO' monad.
lookupSequence_IO :: SequenceData -> IO (Maybe OEISSequence)
lookupSequence_IO :: SequenceData -> IO (Maybe OEISSequence)
lookupSequence_IO SequenceData
x = [OEISSequence] -> Maybe OEISSequence
forall a. [a] -> Maybe a
listToMaybe ([OEISSequence] -> Maybe OEISSequence)
-> IO [OEISSequence] -> IO (Maybe OEISSequence)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> SequenceData -> IO [OEISSequence]
lookupSequences_IO SequenceData
x

-- | Similar to 'lookupSequence_IO', but return up to 10 results.
lookupSequences_IO :: SequenceData -> IO [OEISSequence]
lookupSequences_IO :: SequenceData -> IO [OEISSequence]
lookupSequences_IO = (SequenceData -> String) -> SequenceData -> IO [OEISSequence]
forall a. (a -> String) -> a -> IO [OEISSequence]
getOEIS SequenceData -> String
seqSearchURI

--------------------------------------------------------------------------------

{- $sample

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.
-}

--------------------------------------------------------------------------------

{-# ANN module "HLint: ignore Use camelCase" #-}