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 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)

## Hints for Using On-Line Encyclopedia of Integer Sequences (or OEIS)

• The main uses for the OEIS are:
• to identify a sequence
• to obtain more terms, formulas, references, links, programs, etc. for a sequence

You are trying to solve a problem, but it's too hard.
The answers in the first few cases are 1,1,2,4,9,21,51,127.
You then look this up in the OEIS and see if anyone has ever encountered this sequence before (they have).

Give up?

• Hints:
• Enter about 6 terms, starting with the second term.
Leave off the first term or two, because people may disagree about where the sequence begins.
If there are several 0's and 1's at the start, leave them off too when you are checking to see if your sequence is already in the OEIS.
Don't enter too many terms, because you may have more terms than are in the OEIS.

For the example above, you should enter 2,4,9,21,51,127 and leave off the two 1's.

• Clean up your sequence first.
• If every other term is zero (e.g. 1 0 2 0 4 0 8 0 16 0 32 0 64 0 ...), omit these zeros, and look up 1 2 4 8 16 32 64 instead.
• If your sequence has an obvious common factor (e.g. 2 6 12 20 30 42 ...) try dividing it out and look up 1 3 6 10 15 21 instead.
• In general, do not submit a sequence which is a small constant times an existing sequence

• Examples of searches.
```        1,4,9,16,25,36,64
5 8 13 233 39088169
"fermat's little theorem"
author:Guy keyword:nice
keyword:nice keyword:more -keyword:base
keyword:new -keyword:base
id:A64413
A64413
id:A028284|id:A066948
author:smith keyword:less|keyword:dumb
```
• The search text is a list of space-separated search terms, as with popular web search engines.
• The search terms can be single numbers, sequences of numbers, words or strings of words.
• Sequences of numbers should be separated with commas.
• To search the sequence terms, separate consecutive terms with commas. To search for numbers anywhere in the sequence entry (data, comments, examples etc.), precede the number with a # character and it will be treated as a word (see below).
• Strings of words should be separated by spaces (NOT commas).
• Strings of words may be enclosed in double quotes.
• You can separate search terms with | (no spaces around the |) and it means sequences that match either term.

• Commas and spaces
These three searches produce the same results:
```        1,4,9,16,25,36,49,64,81,100
1 4 9 16 25 36 49 64 81 100
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
```
They both only match sequences in which the numbers appear consecutively, in the order given.
The form with commas is the official form for a sequence search, but the space-only form is used so often that any time three or more numbers appear in a query separated by spaces, the server inserts commas between them automatically. Similarly, the comma-space form is accepted for easier pasting from other sources. All searches will match the squares (A000290).
To match sequences containing those numbers in any order, the numbers must be individually quoted, as in:
```        "1" "4" "9" "16" "25" "36" "49" "64" "81" "100"
```
This search will still match the squares but also other sequences, such as the composite numbers (A002808), which contain all those numbers but not necessarily consecutively or in the given order.

Commas are only inserted for numeric searches, not word searches. Suppose you want to find sequences that cite Stanley's book Enumerative Combinatorics.
The search string Stanley, Combinatorics (with the comma) gets no hits, but searching for Stanley Combinatorics (without the comma) gets several hundred hits.

• Prefixes.
Simple search terms will be matched against every field of the entry,
though fields such as the name and the sequence data are given more weight than others.
Prefixes restrict the search to particular lines, as in:
```        keyword:nice author:LeBrun
```
The prefixes mostly correspond to the lines in the display, as follows:
```        id:                     ref:                    program:
signed:                 formula:                keyword:
name:                   example:                author:
offset:                 maple:                  extension:
comment:                mathematica:
```
• id:A001006 displays sequence A001006. A001006 displays sequence A001006 followed by all other sequences that mention it.
• seq: matches the sequence data, possibly by ignoring signs.
• signed: matches sequence data and requires that any signs match as well. For example
```        signed:1,-1,-1,0,-1,1,-1,0,0,1,-1,0,-1,1,1,0
```
will find the Moebius function mu(n), A008683.
• Put a minus sign in front of a prefix to exclude those matches.
Put a tilde (~) in front of a number to exclude it (only applies if the numbers are separated by spaces).
For example, a search for 2 4 ~8 16 returns sequences such as 0, 1, 2, 4, 16, 65536 that do not contain 8.
• You can say prefix: by itself to match sequences that have a line of that type.
For example ref: will match all sequences with reference lines.
• Another example: you can say, e.g. -seq:5 in your search to exclude all sequences containing the number 5. So if you know your sequence consists only of 1, 2, 3, and 4 (and does include all of them), you could search seq:1 seq:2 seq:3 seq:4 -seq:5 -seq:6 -seq:0.

Two other prefixes are concerned with subsequences:

```        subseq:                 signedsubseq:
```
• You can say, for example, subseq:377,8 to get those numbers in sequence data in that order.
• Relevance.
The results are returned in order of “relevance,” which is a rough measure of how likely you are to want to see a particular sequence.
It orders first by how well the sequence matched the search terms,
second by the popularity of the sequence in other sequences' cross-references,
and finally by sequence number.
Other sort orders can be selected on the results page.

• Wild cards.
• The wildcard _ will match any number in a sequence, as in 1,2,_,4,5
• The wildcard __ will match any list of numbers in a sequence, as in 1,2,__,7,8
• Search terms are highlighted in the results
• Punctuation is not indexed and is treated as identical to whitespace during matching;
for example, searching for fermat's is the same as searching for "fermat s".

• Further examples
• To find sequences containing both terms 1759 and 1957: search for "seq:1759 seq:1957".

• Further Hints
• You can only look up a sequence of integers or whole numbers, like 6 or 10 or -3
• If you have a sequence of fractions, look up numerators and denominators separately
Example: for

1/4, 5/8, 13/16, 29/32, 61/64, 125/128, ...,

look up   1,5,13,29,61,125   and   4,8,16,32,64,128   separately.
• If your sequence has decimal fractions in it (1.4, 5.87, 12.083), the OEIS probably can't help you
(though you might try rounding them to the nearest integer, or taking the integer part).
• There are no letter sequences (such as A, B, C, D ...) in the OEIS!

• For a triangle of numbers, such as:
```           1
1   1
1   2   2
1   3   5   4
1   4   9  12   9
1   5  14  25  30  21
```
etc., make it into a sequence by reading across the rows:

1 2 2   1 3 5 4   1 4 9

and look that up. Or look up the diagonals, such as

1 2 5 12 30 76

• For a square array, such as
```    1   1   1   1 ...
1   2   3   4 ...
2   5   9  14 ...
4  12  25  44 ...
9  30  69 133 ...
21  76 189 392 ...
```
etc., make it into a sequence by reading up (or down) the antidiagonals:

1   1 1   1 2 2   1 3 5 4   1 4 9

and look that up. Or look up the rows or columns, such as

1 2 5 12 30 76

• For a decimal constant, such as 3.14159, look up the sequence of digits, as in

3 1 4 1 5 9

The OEIS search engine will convert numbers containing decimal points into digit sequences automatically,
so you can type 3.14159 directly.
• To search for a single large number in the OEIS, try Google, because Google has searched all the .txt files in the OEIS, and so may do a better job than the OEIS search engine. Example: The 64th Motzkin number is 9468017265749942384739441267, and Google will tell you it is in the b-file for A001006.
• The standard reply returns nicely reformatted sequences.

To see the sequences in the internal format used in the OEIS, click on the "internal format" link on the results page.
This is useful when reporting updates or corrections.

• Note that there is also an Index to the OEIS.
Hint: To search the Index, ask your favorite search engine, e.g. Google, to search for (say) theta series Index OEIS.

• Dates have been standardized in the format Jan 01 2001. The month is one of
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
The day is one of 01 02 03 ... 31, and there may be a comma after the day:
Jan 01 2001 and Jan 01, 2001 are both acceptable.

So to find the sequences submitted or updated on a particular day, look up both "Oct 10 2001" and "Oct 10, 2000" using the "word" search.

• Every sequence in the OEIS has an A-number, such as A001850.
If you know the A-number you can retrieve the sequence directly by typing it (with the A) into the search page.

• If no match is found for your sequence, try Superseeker.

• If your sequence isn't in the database - and is of general interest - please send it to the OEIS.
This will help other people who come across the same sequence (and will preserve your name for posterity).

• For a description of a typical entry in the OEIS, click here.

• To browse the OEIS, use the WebCam. This can be set to browse
• the best sequences
• sequences needing more terms, or
• all sequences.
For the WebCam to work, Javascript must be turned on. It may not work with some older browsers.

• The meaning of the little numbers at the right of the blue bar. For example, +30 and 1086 for A000045.

The +30 is the sequence's query score -- how well it matches the query.
You get 100 points for matching ordering (for example, having 1 3 5 not 1 2 3 4 5 when the query is 1 3 5).
You get 10 points for matching terms in certain lines (for example, sequence data counts more, and sequence number counts a lot more).

The 1086 is the number of sequences in the database that reference the given sequence.

The "relevance" sort is by query score, with ties broken by reference count.

That's how 1 3 5 manages to bring up the odd numbers and 2 3 5 the primes.