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32 33 </head> 34 <body bgcolor=#ffffff onload="redir();sf()"><a href="https://oeis.org/cdn-cgi/content?id=Nv8uCuDMi8MAnJ271uJi0zigWGZya76dgdpLfxaPgIw-1790238591.6008272-1.2.1.1-JhujeyS4FTdcPx1rsm5o7w8SuL5ifqiNF2.b6YGIP6rDB7bKPdrmi7lOaZP.YQgp" aria-hidden="true" rel="nofollow noopener" style="display: none !important; visibility: hidden !important"></a> 35 <div class=loginbar> 36 <div class=login> 37 38 <a href="/login?redirect=%2Fhints.html" rel="nofollow">login</a> 39 40 </div> 41 </div> 42 43 <div class=center><div class=top> 44 <center> 45 <div class=donors> 46 The OEIS is supported by <a href="http://oeisf.org/#DONATE">the many generous donors to the OEIS Foundation</a>. 47 </div> 48 49 <div class=banner> 50 <a href="/"><picture> 51 <source media="(min-width: 830px)" srcset="/oeis-banner-wide.png 1x, /[email protected] 2x" width="918" height="129"> 52 <img class=banner border="0" width="540" height="129" src="/oeis-banner.png" srcset="/oeis-banner.png 1x, /[email protected] 2x" alt="Hints for Using The On-Line Encyclopedia of Integer Sequences"> 53 </picture></a> 54 </div> 55 56 57 58 59 <div class="motdbox" style="justify-content: center;"> 60 <div class="400k"> 61 <p>Nominations for sequence A400000 are open, see <a href="https://oeis.org/wiki/Nominations_for_Sequence_A400000">here</a>.</p> 62 </div> 63 64 </div> 65 66 </center> 67 </div></div> 68 69 <div class=center><div class=pagebody> 70 71 72 <div class=searchbarcenter> 73 <form name=f action="/search" method="GET"> 74 <div class=searchbargreet> 75 <div class=searchbar> 76 <div class=searchq> 77 <input class=searchbox maxLength=1024 name=q value="" title="Search Query"> 78 79 80 81 82 </div> 83 <div class=searchsubmit> 84 <input type=submit value="Search" name=go> 85 </div> 86 <div class=hints> 87 <span class=hints><a href="/hints.html">Hints</a></span> 88 </div> 89 </div> 90 <div class=searchgreet> 91 (Greetings from <a href="/welcome">The On-Line Encyclopedia of Integer Sequences</a>!) 92 </div> 93 </div> 94 </form> 95 </div> 96 97<div align="center"> 98<h2>Hints for Using <a href="https://oeis.org/">On-Line 99Encyclopedia of Integer Sequences</a> (or OEIS)</h2> 100</div> 101 102 103<ul> 104<li><STRONG>The main uses for the OEIS are:</STRONG> 105<ul> 106<li>to identify a sequence</li> 107<li>to obtain more terms, formulas, references, links, programs, etc. for a sequence</li> 108</ul> 109<p> 110You are trying to solve a problem, but it's too hard.<br> 111The answers in the first few cases are 112<A HREF="A001006">1,1,2,4,9,21,51,127</a>.<br> 113You then look this up</a> in the OEIS and see if anyone 114has ever encountered this sequence before (they have). 115<p> 116<center> 117<a href="https://oeis.org/search?q=1%2C1%2C2%2C4%2C9%2C21%2C51%2C127&sort=&language=&go=Search">Give up?</a> 118<form action="/search"> 119 <input maxLength=1024 size=55 name=q value="2,4,9,21,51,127" 120 title="Search Query"><input type=submit value="Search" name=go> 121</form> 122</center> 123<p> 124</li> 125 126<li><strong>Hints:</strong> 127<ul> 128<li><font color="red"><strong>Enter about 6 terms, starting with the second term.</strong></font><br> 129Leave off the first term or two, because people may disagree about where 130the sequence begins.<br> 131If 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.<br> 132Don't enter too many terms, because you may have more terms than are in the OEIS.<p> 133 For the example above, you should enter 2,4,9,21,51,127 and 134leave off the two 1's.<p> 135</li> 136 137<li><STRONG>Clean up your sequence first.</STRONG> 138<ul> 139<li>If every other term is zero (e.g. 1401 0 2 0 4 0 8 0 16 0 32 0 64 0 ...), omit these zeros, 141and look up 1 2 4 8 16 32 64 instead.</li> 142<li>If your sequence has an obvious common factor 143(e.g. 2 6 12 20 30 42 ...) try dividing it out 144and look up 1 3 6 10 15 21 instead.</li> 145<li>In general, do not submit a sequence which is a small constant times an existing sequence</li> 146</ul> 147<p> 148</li> 149</ul> 150</li> 151 152 153<li> 154<STRONG>Advanced search syntax</STRONG> 155 156<center> 157<form action="/search"> 158 <input maxLength=1024 size=55 name=q value="2,3,6,16 "symmetric group" author:Stanley" 159 title="Search Query"><input type=submit value="Search" name=go> 160</form> 161</center> 162 163<ul> 164<li> 165<strong><font color="red">Examples of searches.</font></strong> 166 167<pre> 168 1,4,9,16,25,36,64 169 5 8 13 233 39088169 170 "fermat's little theorem" 171 author:Guy keyword:nice 172 keyword:nice keyword:more -keyword:base 173 keyword:new -keyword:base 174 id:A64413 175 A64413 176 id:A028284|id:A066948 177 author:smith keyword:less|keyword:dumb 178</pre> 179 180<ul> 181<li>The search text is a list of space-separated search terms, 182as with popular web search engines.</li> 183<li>The search terms can be single numbers, 184sequences of numbers, words or strings of words.</li> 185<li>Sequences of numbers should be separated with commas.</li> 186<li>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).</li> 187<li>Strings of words should be separated by spaces (NOT commas).</li> 188<li>Strings of words may be enclosed in double quotes.</li> 189<li>You can separate search terms with | (no spaces around
190the |) and it means sequences that match either term.</li> 191 192</ul> 193 194<p> 195</li> 196 197<li> 198<strong><font color="red">Commas and spaces</font></strong><br> 199 200These three searches produce the same results: 201 202<pre> 203 1,4,9,16,25,36,49,64,81,100 204 1 4 9 16 25 36 49 64 81 100 205 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 206</pre> 207 208They both only match sequences in which the numbers appear consecutively, 209in the order given.<br> 210The form with commas is the official form for a sequence search, 211but the space-only form is used so often that 212any time three or more numbers appear 213in a query separated by spaces, the server inserts commas between them automatically. 214Similarly, the comma-space form is accepted for easier pasting from other sources. 215All searches will match the squares (A000290). 216<br> 217To match sequences containing those numbers in any order, 218the numbers must be individually quoted, as in: 219 220<pre> 221 "1" "4" "9" "16" "25" "36" "49" "64" "81" "100" 222</pre> 223 224This search will still match the squares but also 225other sequences, 226 such as the composite numbers (A002808), 227which contain all those numbers but not necessarily consecutively or 228in the given order.<p> 229 230Commas are only inserted for numeric searches, not word searches. 231Suppose you want to find sequences that cite Stanley's book Enumerative Combinatorics.<br> 232The search string Stanley, Combinatorics (with the comma) gets no hits, but searching for Stanley Combinatorics (without the comma) gets several hundred hits. 233<p> 234</li> 235 236<li> 237<strong><font color="red">Prefixes.</font></strong><br> 238Simple search terms will be matched against every field of the entry,<br> 239though fields such as the name and the sequence data are given 240more weight than others.<br> 241Prefixes restrict the search to particular lines, as in: 242 243<pre> 244 keyword:nice author:LeBrun 245</pre> 246 247The prefixes mostly correspond to the lines in the display, as follows: 248 249<pre> 250 id: ref: program: 251 seq: link: xref: 252 signed: formula: keyword: 253 name: example: author: 254 offset: maple: extension: 255 comment: mathematica: 256</pre> 257 258<ul> 259<li><tt>id:A001006</tt> displays sequence A001006. 260<tt>A001006</tt> displays sequence A001006 followed by all other sequences 261that mention it.</li> 262<li><tt>seq:</tt> matches the sequence data, possibly by ignoring signs.</li> 263<li><tt>signed:</tt> matches sequence data and requires that any signs match 264as well. For example 265<pre> 266 signed:1,-1,-1,0,-1,1,-1,0,0,1,-1,0,-1,1,1,0 267</pre> 268will find the Moebius function mu(n), A008683. 269</li> 270<li>Put a minus sign in front of a prefix to exclude those matches.<br> 271Put a tilde (~) in front of a number to exclude it (only applies 272if the numbers are separated by spaces).<br> 273For example, a search for 2 4 ~8 16 returns sequences such as 0, 1, 2, 4, 16, 65536 274that do not contain 8. 275</li> 276<li>You can say prefix: by itself to match sequences 277that have a line of that type.<br> 278For example <tt>ref:</tt> 279will match all sequences with reference lines. 280</li> 281<li> 282Another example: 283you can say, e.g. -seq:5 in your search to exclude all 284sequences containing the number 5. So if you know your sequence 285consists only of 1, 2, 3, and 4 (and does include all of them), you 286could search seq:1 seq:2 seq:3 seq:4 -seq:5 -seq:6 -seq:0. 287</li> 288</ul> 289<p> 290 291Two other prefixes are concerned with subsequences: 292 293<pre> 294 subseq: signedsubseq: 295</pre> 296 297<ul> 298<li> 299You can say, for example, subseq:377,8 to get those numbers in sequence data in that order. 300</li> 301<p> 302</ul> 303 304 305</li> 306 307<li> 308<strong><font color="red">Relevance.</font></strong><br> 309The results are returned in order of “relevance,” which is a rough 310measure of how likely you are to want to see a particular 311sequence.<br> 312It orders first by how well the sequence matched the search 313terms,<br> second by the popularity of the sequence in other 314sequences' cross-references,<br> and finally by 315sequence number.<br> 316Other sort orders can be selected on the results page. 317<p> 318</li> 319 320<li> 321<strong><font color="red">Wild cards.</font></strong><br> 322<ul> 323<li> 324 The wildcard _ will match any number in a sequence, 325 as in 1,2,_,4,5 326</li> 327 328<li> 329 The wildcard __ will match any list of numbers in a sequence, 330 as in 1,2,__,7,8 331</li> 332 333<li> 334 Search terms are highlighted in the results 335</li> 336 337</ul> 338 339</li> 340<p> 341 342<li> 343<strong><font color="red">Punctuation</font></strong>
344is not indexed and is treated as identical to whitespace 345during matching;<br> 346for example, 347searching for <tt>fermat's</tt> is the same as searching for <tt>"fermat s"</tt>. 348<p> 349</li> 350 351<li> 352<strong><font color="red">Further examples</font></strong> 353<ul> 354 355 <li> 356 To find sequences containing both terms 1759 and 1957: 357 search for <tt>"seq:1759 seq:1957"</tt>. 358 </li> 359 360</ul> 361<p> 362</li> 363 364</ul> 365 366<p> 367</li> 368 369 370 371<li> 372<strong>Further Hints</strong> 373<ul> 374<li><STRONG><A NAME="fract">You can only look up a sequence of integers or whole numbers, 375like 6 or 10 or -3</A></STRONG> 376<ul> 377<li>If you have a sequence of fractions, <strong>look up numerators 378and denominators separately</strong><br> 379Example: for 380<p> 1/4, 5/8, 13/16, 29/32, 61/64, 125/128, ...,</p> 381look up 1,5,13,29,61,125 382and 4,8,16,32,64,128 separately.</li> 383<li>If your sequence has <strong>decimal fractions</strong> in it (1.4, 5.87, 12.083), 384the OEIS probably can't help you<br> 385(though you might try rounding them to the nearest integer, 386or taking the integer part).</li> 387<li>There are <strong>no letter sequences</strong> (such as A, B, C, D ...) 388in the OEIS!</li> 389</ul> 390<p> 391</li> 392 393<li><font color="green"><strong>For a triangle of numbers,</strong></font> such as: 394<pre> 395 1 396 1 1 397 1 2 2 398 1 3 5 4 399 1 4 9 12 9 400 1 5 14 25 30 21 401</pre> 402 403etc., make it into a sequence by reading across the rows: 404<p> 1 2 2 1 3 5 4 1 4 9</p> 405and look that up. Or look up the diagonals, such as 406<p> 1 2 5 12 30 76</p> 407<p> 408</li> 409<li><font color="green"><strong>For a square array,</strong></font> such as 410<pre> 411 1 1 1 1 ... 412 1 2 3 4 ... 413 2 5 9 14 ... 414 4 12 25 44 ... 415 9 30 69 133 ... 416 21 76 189 392 ... 417</pre> 418etc., make it into a sequence by reading up (or down) the <strong>antidiagonals</strong>: 419<p> 1 1 1 1 2 2 1 3 5 4 1 4 9</p>
420and look that up. Or look up the rows or columns, such as 421<p> 1 2 5 12 30 76</p> 422<p> 423</li> 424 425<li><font color="green"><strong>For a decimal constant,</strong></font> such as 4263.14159, look up the sequence of digits, as in 427<p> 3 1 4 1 5 9</p> 428The OEIS search engine will convert numbers containing decimal points 429into digit sequences automatically,<br> so you can type 3.14159 directly. 430</li> 431<p> 432 433 434<li> 435To search <font color="green">for <strong>a single large number</strong></font> in 436the 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. 437Example: The 64th Motzkin number is 9468017265749942384739441267, and Google will tell you it is in the <a href="https://oeis.org/A001006/b001006.txt">b-file</a> for <a href="A001006">A001006</a>. 438</li> 439<p> 440 441 442 443</ul> 444</li> 445 446 447 448 449<li>The <a href="https://oeis.org/"><STRONG>standard reply</STRONG></a> 450returns nicely reformatted sequences.<p> 451To see the sequences in the <STRONG>internal format</STRONG> used in the OEIS, 452click on the "internal format" link on the results page.<br> 453This is useful when reporting updates or corrections. 454<p> 455</li> 456 457<li> 458Note that there is also an 459<a href=Sindx.html> 460<STRONG>Index to the OEIS</STRONG></a>. 461<br> 462Hint: To search the <a href=Sindx.html>Index</a>, 463ask your favorite search engine, e.g. Google, 464to search for (say) theta series Index OEIS. 465<p> 466</li> 467 468<p> 469 470</li> 471 472 473<li><STRONG>Dates</STRONG> have been standardized in the format Jan 01 2001. The month 474is one of<br> 475 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec<br> 476The day is one of 01 02 03 ... 31, and there may be a comma after the day:<br> 477Jan 01 2001 and Jan 01, 2001 are both acceptable.<p> 478So to find the sequences submitted or updated on a particular day, 479look up both "Oct 10 2001" and "Oct 10, 2000" using the "word" search. 480<p> 481</li> 482 483<li>Every sequence in the OEIS has an <strong>A-number</strong>, such as 484<A HREF="A001850">A001850</a>.<br> 485If you know the A-number you can retrieve the sequence directly by typing it (with the A) into the search page. 486<p> 487</li> 488 489<li>If no match is found for your sequence, try 490<a href="ol.html"><STRONG>Superseeker</STRONG></a>. 491<p> 492</li> 493 494<li><STRONG>If your sequence isn't in the database</STRONG> - and is 495of general interest - please 496<a href="Submit.html"><STRONG>send it to the OEIS</STRONG></a>.<br> 497This will help other people who come across the same sequence 498(and will preserve your name for posterity). 499<p> 500 501</li> 502<li><STRONG>For a description of a typical entry in the OEIS,</STRONG> click 503<a href="eishelp2.html"><STRONG>here</STRONG></a>. 504<p> 505</li> 506 507<li> 508<STRONG>To browse the OEIS,</STRONG> use the 509<a href="WebCam.html"><STRONG>WebCam</STRONG></a>. 510This can be set to browse 511<ul> 512<li>the best sequences</li> 513<li>recent additions</li> 514<li>sequences needing more terms, or</li> 515<li>all sequences.</li> 516</ul> 517<small>For the WebCam to work, Javascript must be turned on. It may not work with some older browsers.</small> 518<p> 519</li> 520 521<li> 522<strong>The meaning of the little numbers at the right of the blue bar.</strong> 523For example, +30 and 1086 for A000045. 524<p> 525The +30 is the sequence's query score -- how well it matches the query.<br> 526You get 100 points for matching ordering (for example, 527having 1 3 5 not 1 2 3 4 5 when the query is 1 3 5).<br> 528You get 10 points 529for matching terms in certain lines (for example, sequence data 530counts more, and sequence number counts a lot more). 531<p> 532The 1086 is the number of sequences in the database that reference 533the given sequence. 534<p> 535The "relevance" sort is by query score, with ties broken by reference count. 536<p> 537That's how 1 3 5 manages to bring up the odd numbers and 2 3 5 the primes. 538<p> 539</li> 540 541<li> 542<font color="red"><STRONG>For more information, see:</STRONG></font> 543<ul> 544<li> <a href="https://oeis.org/wiki/Welcome"><STRONG>Welcome to the On-Line Encyclopedia.</STRONG></a> </li> 545<li> <font color="red"><strong>My Favorite Integer Sequences</strong></font> 546[<a href="http://NeilSloane.com/doc/sg.txt">Abstract</a>, 547<a href="http://NeilSloane.com/doc/sg.pdf">pdf</a>, 548<a href="http://NeilSloane.com/doc/sg.ps">ps</a>].</li> 549 550<li> An article about the OEIS 551from the <strong>Handbook of Computer Science</strong> 552[<a href="http://NeilSloane.com/doc/hbk.pdf">pdf</a>, 553<a href="http://NeilSloane.com/doc/hbk.ps">
553ps</a>].</li> 554<li> 555B. Hayes, 556<a href="http://www.americanscientist.org/template/AssetDetail/assetid/21436">A 557Question of Numbers</a>, in <STRONG>American Scientist</STRONG>. 558</li> 559</ul> 560</li> 561 562</ul> 563 564 565 </div></div> 566 567 <p> 568 569 <div class=footerpad></div> 570 <div class=footer> 571 <center> 572 <div class=bottom> 573 <div class=linksbar> 574 <a href="/">Lookup</a> 575 <a href="/wiki/Welcome"><font color="red">Welcome</font></a> 576 <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> 577 <a href="/account">Register</a> 578 579 <a href="/play.html">Music</a> 580 <a href="/plot2.html">Plot 2</a> 581 <a href="/demo1.html">Demos</a> 582 <a href="/wiki/Index_to_OEIS">Index</a> 583 <a href="/webcam">WebCam</a> 584 <a href="/Submit.html">Contribute</a> 585 <a href="/eishelp2.html">Format</a> 586 <a href="/wiki/Style_Sheet">Style Sheet</a> 587 <a href="/transforms.html">Transforms</a> 588 <a href="/ol.html">Superseeker</a> 589 <a href="/recent">Recents</a> 590 </div> 591 <div class=linksbar> 592 <a href="/community.html">The OEIS Community</a> 593 </div> 594 <div class=linksbar> 595 Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> 596 </div> 597 598 <div class=dbinfo>Last modified September 24 04:27 EDT 2026. Contains 399566 sequences.</div> 599 <div class=legal> 600 <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> 601 </div> 602 603 </div> 604 </center> 605 </div> 606
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Line numbers count LF bytes from the start of the resource, as the search results do. Vendor segments are library code the classifier recognised; they are stored but not indexed. Bytes are shown as Latin1 characters, one per byte.