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."]
*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.

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

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.