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Identifying a Sequence: Supplying a Formula

A secondary goal of the OEIS is to provide a place where the general public has access to interesting parts of mathematics.

Suppose someone rediscovers the sequence of tetrahedral numbers, the number of balls in a triangular pyramid, shown here:

The first few numbers are easy to calculate by hand:

1, 4, 10, 20, 35, 56, ...

This person might be a high-school student in Tokyo, a medical doctor in Paris, or a retired mountain climber in South Dakota. He or she would like to know if there is a formula for these numbers, what they are called, and a reference where they can find out more about them.

As long as they have access to the Internet or to electronic mail, they can consult the OEIS. (If they don't have access to either the Internet or email, even if they do not have electricity - like the correspondent in South Dakota - they can still refer to the book version, published in 1995 by Academic Press. This is now out of date, but includes some 5000 of the most important sequences.)

For the moment, let us suppose they can access the Internet. (Consulting the database via email will be discussed in a later demonstration.) They go to the main web page, where they see the following.

The On-Line Encyclopedia of Integer Sequences

Enter a sequence, word, or sequence number:
You replace the example by your sequence and click "Submit":

The reply shows several sequences that match these terms, but the top entry is the sequence that is sought:

Greetings from the On-Line Encyclopedia of Integer Sequences!

 A000292 Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6. (Formerly M3382 N1363) 326
 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, 6545, 7140, 7770, 8436, 9139, 9880, 10660, 11480, 12341, 13244, 14190, 15180 (list; graph; refs; listen; history; edit; internal format)

The reply gives more terms, the name of the sequence, a formula for the nth term, a generating function, and several references and links where they can find out more about the sequence.

The Beiler reference in particular (a wonderful book) has lured many people into studying mathematics for pleasure.

No doubt the new book by Conway and Guy (also highly recommended for general readers) will accomplish the same thing.

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Last modified August 14 00:08 EDT 2024. Contains 375146 sequences. (Running on oeis4.)