OEIS, the On-Line Encyclopedia of Integer Sequences, is probably the authoritative source for information on integer sequences, which arise primarily from combinatorics, number theory, and recreational mathematics. However, the often deep connections between mathematical patterns and those in the real world mean that numerous sequences can be identified in terms of physical or chemical phenomena, and most branches of mathematics are represented in some fashion.
It is easily the largest such collection available, anywhere. As of 2004, there are over 100,000 entries in the database, which can be queried by web-search (at http://www.research.att.com/~njas/sequences) or via an email service (details at http://www.research.att.com/~njas/sequences/ol.html). As the URLs suggest, the project is hosted by AT&T research; however, it is largely the product of 40 years of collecting by mathematician Neil Sloane. Two books of his collection have been released, in 1973 and 1995; but the subsequent influx of sequences has rendered the project unmanageable as a printed work and since 1996 the complete database has existed only as a (free) internet resource, maintained primarily by Sloane but supported in recent years by volunteers and a board of editors. The site grows at a rate of about 10,000 sequences a year.
An entry for a sequence offers far more than just a name and some terms. A stated aim of the OEIS is to benefit active researchers in the sciences and mathematics, both by simplifying work through supplying general formulae or lists of terms and by highlighting previous work on a given sequence through journal articles, books, websites and related sequences. Each sequence has an ID number and URL on the OEIS for ease of reference and citation.
The standard search process is to simply drop in the first half dozen or so terms into the web search page. However, advanced searching on the text of the entries is possible (to find work by a given author or in a certain field); and there is a super-seeker that tries particularly hard to match sequences by applying various transformations to the supplied terms, although due to the additional computational overhead it is requested that its usage be kept to a minimum.
In addition to being a crucial academic resource, the database is a goldmine for curious mathematical diversions. It's possible to browse for random entries, drawn either from the entire database or the "best" highlights. A puzzles page challenges you to determine the rule behind a given collection of terms. The "hot" page picks out some of the best and worst recent additions- which can also be explored with the 'webcam', which serves up recent contributions every few seconds until you spot one you like... as the frontpage observes, all this can become strangely addictive...