Skip to content
Puzzlehead
The “mathematical equivalent to the FBI’s voluminous fingerprint files” turns 50 this year, with 362,765 entries (and counting).
The Fibonacci sequence, in which every term (starting with the 3rd term) is the sum of the two preceding numbers, is one of the more familiar sequences in the On-Line Encyclopedia of Integer Sequences, which first appeared as a handbook in 1973.Credit...Tony Cenicola/The New York Times
Supported by
Continue reading the main story
Send any friend a story
As a subscriber, you have 10 gift articles to give each month. Anyone can read what you share.
163
(Video) What Number Comes Next? - Numberphile
By Siobhan Roberts
Some numbers are odd:
1, 3, 5, 7, 9, 11, 13, 15 …
Some are even:
2, 4, 6, 8, 10, 12, 14, 16 …
And then there are the puzzling “eban” numbers:
2, 4, 6, 30, 32, 34, 36, 40 …
What number comes next? And why?
These are questions that Neil Sloane, a mathematician of Highland Park, N.J., loves to ask. Dr. Sloane is the founder of the On-Line Encyclopedia of Integer Sequences, a database of 362,765 (and counting) number sequences defined by a precise rule or property. Such as the prime numbers:
2, 3, 5, 7, 11, 13, 17, 19 …
Or the Fibonacci numbers — every term (starting with the 3rd term) is the sum of the two preceding numbers:
0, 1, 1, 2, 3, 5, 8, 13 …
This year the OEIS, which has been praised as “the master index to mathematics” and “a mathematical equivalent to the FBI’s voluminous fingerprint files,” celebrates its 50th anniversary. The original collection, “A Handbook of Integer Sequences,” appeared in 1973 and contained 2,372 entries. In 1995, it became an “encyclopedia,” with 5,487 sequences and an additional author, Simon Plouffe, a mathematician in Quebec. A year later, the collection had doubled in size again, so Dr. Sloane put it on the internet.
“In a sense, every sequence is a puzzle,” Dr. Sloane said in a recent interview. He added that the puzzle aspect is incidental to the database’s main purpose: to organize all mathematical knowledge.
Sequences found in the wild — in mathematics, but also quantum physics, genetics, communications, astronomy and elsewhere — can be puzzling for numerous reasons. Looking up these entities in the OEIS, or adding them to the database, sometimes leads to enlightenment and discovery.
“It’s a source of unexpected results,” said Lara Pudwell, a mathematician at Valparaiso University in Indiana and a member of the OEIS Foundation’s board of trustees. Dr. Pudwell writes algorithms to solve counting problems. A few years ago, thus engaged, she entered into the OEIS search box a sequence that arose while studying numerical patterns:
2, 4, 12, 20, 38, 56, 88 …
The only result that popped up pertained to chemistry: specifically, to the periodic table and the atomic numbers of the alkaline earth metals. “I found this perplexing,” Dr. Pudwell said. She consulted with chemists and soon “realized there were interesting chemical structures to work with to explain the connection.”
Sequence serendipity provides what Russ Cox, a software engineer at Google, called “amazing cross-connective tissue for the sciences.” Dr. Cox, based in Cambridge, Mass., is the president of the OEIS board. He submitted his first sequence, which emerged from a programming contest puzzle, as a high-school student in 1996. He has twice rewritten the software for the database, which he thinks of as “the collective wisdom of math and science in this interesting numerical form.”
Donald Knuth, a computer scientist at Stanford, known for his analysis of algorithms, among other things, has also chanced upon breakthroughs. Working on a new problem, he always searches the OEIS. “It finds my bedfellows,” he said. “The beautiful thing is that you can compute your way into the literature.”
Image
A numerical punchline
Searching the database for “puzzle” produces more than 1,700 results. Some entries describe physical puzzles, such as the Rubik’s cube and its “God’s Number,” the maximum number of turns needed to solve a scrambled cube: 11 for a 2 × 2 × 2 cube, 20 for the classic 3 × 3 × 3 cube. Beyond that — say, for the 4 × 4 × 4 Rubik’s Revenge — God’s Number is uncertain, making for a short sequence thus far: 0, 11, 20.
“We have bounds on God’s number for 4 × 4 × 4, but they’re very loose,” Erik Demaine, a computer scientist at the Massachusetts Institute of Technology, said in an email. “We know it’s between 31 and 16,777,214.”
The OEIS is long on sequences that tease and land like a joke — like the eban sequence, invented by Dr. Sloane. “It’s very, very simple,” he said. “And yet nobody ever guesses it.”
More terms are not necessarily helpful: 2, 4, 6, 30, 32, 34, 36, 40, 42, 44, 46, 50, 52, 54, 56, 60, 62, 64, 66, 2000, 2002, 2004, 2006, 2030 …
The punchline of the eban numbers is that the letter “e” is banned: This sequence contains no numbers that, when the numbers are spelled out, contain the letter “e.”
“Eban sounds a bit like ‘even,’ and it’s a very nice pun,” Dr. Sloane said — because if you look at numbers without an “e,” they are all even. “This is an old theorem of mine: that every odd number has an ‘e’ in it, in English. And so all the numbers where ‘e’ is banned are even. Of course, not all even numbers are eban, but enough are to make a good sequence.”
What’s the next number?
Here are a few more sequences to try (answers below):
1) 0, 1, 8, 11, 69, 88, 96, 101 …
2) 1, 11, 21, 1211, 111221, 312211 …
3) 5, 8, 12, 18, 24, 30, 36, 42 …
4) 14, 18, 23, 28, 34, 42, 50 …
Dr. Sloane first went looking for a sequence in 1964, as a graduate student. Studying paths in an artificial neural network (a mathematical system mimicking the human brain), his computations generated:
0, 1, 8, 78, 944, 13800 …
“I badly needed a formula for the nth term in order to determine the rate of growth of the terms,” he wrote in a retrospective published in April. “This would indicate how long the activity in this very simple neural network would persist.” Hunting through textbooks, reference books, journals, he came close, but no sequence. Eventually, together with the combinatorialist John Riordan, he figured out the formula, and the next term: 237432.
Along the way, Dr. Sloane recorded sequences on file cards, then on punched cards. “These were never called ‘punch cards’,” he wrote. “To anyone who worked with them in the 1960s, ‘punch cards’ sounds like ‘grill cheese’ for ‘grilled cheese.’”
Once online in 1995, it dominated his home page on the website of AT&T Labs, where he worked for 43 years. In 2010, it became a moderated wiki and now is run by about 170 international volunteer editors who wrangle 50 or more new submissions a day. The OEIS Foundation recently announced an endowment campaign to fund a full-time managing editor.
Dr. Sloane’s signature lecture on the subject is titled “Confessions of a Sequence Addict.” And he showcases his favorite sequences on Numberphile, an educational math YouTube channel, ever asking “What number comes next?” with the likes of this procession:
1, 4, 8, 48, 88, 488 …
The best way to puzzle it out is to “stare at these numbers,” he advised. “Look at the number 8. What do you see?”
“Well, you see ‘8,’ but more important you see two holes,” he said. “When you look at 48, you see three holes; 88 has four holes in it; 4 has one hole.”
“And 1 has no holes in it. So, the definition of this sequence is: The smallest positive number that has ‘n’ holes in it.” (That is, assuming an enclosed “4” and a “2” with no closed loop.)
Image
A Sisyphean task
Among Dr. Sloane’s current favorites is the Sisyphus sequence, devised in 2022 by Éric Angelini, a journalist and amateur mathematician in Brussels, and his sometime accomplice, Carole Dubois, of Toulouse, France.
1, 3, 6, 3, 8, 4, 2, 1, …
The Sisyphus rule: If the number is even, divide by two. If the number is odd, add the smallest prime number not yet added. The first term, 1, is odd, so adding the smallest prime number, 2, gives 3; 3 is odd, so adding the next prime, 3, gives 6; 6 is even, so dividing by two gives 3, and so on.
“Now here’s the interesting question,” said Dr. Sloane: Some numbers appear more than once, but does every number appear at least once? “We don’t know.”
The number 36 was stubborn — it was still absent after the sequence had been computed to a billion terms. This was “really worrying,” Dr. Sloane said. “It would be like a flaw in the universe, if 36 was missing.”
At the end of an OEIS board meeting last year, Dr. Sloane, the chairman, asked if there was any further business, and since there wasn’t, he explained how Sisyphus was stuck. He asked the assembled experts, “Does 36 ever appear in this beautiful sequence?” Two hours later, Dr. Cox found it at the far reaches of 77 billion terms — the 77,534,485,877th term lands on 36.
While still pondering Sisyphus, Dr. Sloane is also — always — keeping tabs on any number of consequential developments and discoveries. Mr. Angelini called him a “puzzle chaser.” Dr. Sloane admitted to being a “sequence chaser.” And when not in hot pursuit, he is working on his book, “The Joy of Seqs.”
Answers:
1) 111. These are the “strobogrammatic” numbers — the same upside-down, or rotated 180 degrees (not flipped along the horizontal axis).
2) 13112221. This is the “Look and Say” sequence: At the first term, 1, you describe what you see — one 1 — so the next term is 11. At the second term (11) you again describe what you see — two 1s — so the next term is 21. Describing that — one 2 and one 1 — results in the next term, 1211, and so on.
3) 52. The sums of two consecutive primes.
4) 59. The numbered stops on the 1 train in Manhattan.
A version of this article appears in print on , Section
D
, Page
8
of the New York edition
with the headline:
What’s the Next Number? This Database Knows.. Order Reprints | Today’s Paper | Subscribe
163
163
Advertisement
Continue reading the main story
FAQs
What is the on-line encyclopedia of integer sequences? ›
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009. Sloane is the chairman of the OEIS Foundation.
What is the handbook of integer sequences? ›A Handbook of Integer Sequences contains a main table of 2300 sequences of integers that are collected from all branches of mathematics and science. This handbook describes how to use the main table and provides methods for analyzing and describing unknown and important sequences.
What are the 4 types of sequences? ›There are four main types of different sequences you need to know, they are arithmetic sequences, geometric sequences, quadratic sequences and special sequences.
What chapter is integers in? ›| 1 | -10 | 0 |
|---|---|---|
| -4 | -3 | -2 |
| -6 | 4 | -7 |
An integer is a number that includes negative and positive numbers, including zero. It does not include any decimal or fractional part. A few examples of integers are: -5, 0, 1, 5, 8, 97, and 3,043.
What is the next number in the sequence 1 2 4 7 ___ ___ 22? ›Sequence: 1, 2, 4, 7, 11, 16, 22, 29, 37, ...
What is the next number in the sequence 2 3 4 6 6? ›2, 3, 4, 6, 6, 9, 8, 12, 10 -
What's the next number in the sequence 1 1 2 4 3 9 4? ›Solution: Given, sequence 1, 1, 2, 4, 3, 9, 4, …… Hence, the next number is 55.
What are the famous sequences of integers? ›| Name | First elements |
|---|---|
| Fibonacci numbers F(n) | {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...} |
| Sylvester's sequence | {2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, ...} |
| Tribonacci numbers | {0, 1, 1, 2, 4, 7, 13, 24, 44, 81, ...} |
| Powers of 2 | {1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...} |
Step 1: Find the common difference of each pair of consecutive terms in the sequence by subtracting each term from the term that comes directly after it. Step 2: Add the common difference to the number prior to the first missing number in the sequence. Step 3: Repeat Step 2 for any other missing numbers.
What is 2 4 8 16 sequence called? ›
It is a geometric sequence.
What are 5 examples of sequences in math? ›| First Term | Term-to-Term Rule | First 5 Terms |
|---|---|---|
| 3 | Add 6 | 3, 9, 15, 21, 27, … |
| 8 | Subtract 2 | 8, 6, 4, 2, 0, … |
| 12 | Add 7 | 12, 19, 26, 33, 40, … |
| -4 | Subtract 5 | -4, -9, -14, -19, -24, … |
A sequence is an ordered list of numbers . The three dots mean to continue forward in the pattern established. Each number in the sequence is called a term. In the sequence 1, 3, 5, 7, 9, …, 1 is the first term, 3 is the second term, 5 is the third term, and so on.
What grade level is integers? ›If you are in a state that teaches Common Core, 6th grade is when the conceptual understanding of integers begins. Students are expected to be able to place and order integers on a number line, as well as determine opposites (6.
What are the first 12 integers? ›First 12 natural numbers are: 1,2,3,4,5,6…. 12. Now, substitute 78 for the sum of observations and 12 for the total number of observations in the formula $M = \dfrac{S}{n}$ to determine the arithmetic mean. Hence, 6.5 is the arithmetic mean of the first 12 natural numbers.
What are the 7 integers? ›The examples of integers are, 1, 2, 5,8, -9, -12, etc. The symbol of integers is “Z“.
What are the 4 rules of integers? ›RULE 1: The product of a positive integer and a negative integer is negative. RULE 2: The product of two positive integers is positive. RULE 3: The product of two negative integers is positive. RULE 1: The quotient of a positive integer and a negative integer is negative.
What are the 3 sets of integers? ›Integers are sometimes split into 3 subsets, Z+, Z- and 0. Z+ is the set of all positive integers (1, 2, 3, ...), while Z- is the set of all negative integers (..., -3, -2, -1). Zero is not included in either of these sets .
What are 5 examples of integers numbers? ›An integer (pronounced IN-tuh-jer) is a whole number (not a fractional number) that can be positive, negative, or zero. Examples of integers are: -5, 1, 5, 8, 97, and 3,043. Examples of numbers that are not integers are: -1.43, 1 3/4, 3.14, .09, and 5,643.1.
What is the sequence 1 2 4 7 12? ›1, 2, 4, 7, 12, 20, 33, 54, 88, ... with offset 1. This sequence counts the number of Fibonacci meanders. A Fibonacci meander is a meander which does not change direction to the left except at the beginning of the curve where it is allowed to make (or not to make) as many left turns as it likes.
What is the sequence 1 1 2 4 7 11? ›
1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, ... Its three-dimensional analogue is known as the cake numbers.
Which is next in the sequence 1 2 7 14 23? ›Detailed Solution
Here, the missing term is 98.
The missing number is 88-2=86.
What is the next number in the sequence 2 3 5 7 11 13 _______? ›Answer and Explanation: The number that comes next in the series 2, 3, 5, 7, 11, 13 would be 17.
What is the sequence of 2 7 8 3 12? ›Summary: The next number in the sequence 2, 7, 8, 3, 12, 9 is 4.
How can we called the following sequence 1 1 2 3 5 8 13 21? ›The Fibonacci sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34... In this series, the next number is found by adding the two numbers before it. Hence, the next term in the series is 8 + 13 = 21.
What sequence is the sequence 1 2 1 4 1 6 1 8 an example of? ›Sequences with such patterns are called arithmetic sequences.
What is the next number in the sequence 2 3 4 6 8 9 ___? ›2, 3, 4, 4, 6, 8, 9, 12, 16.
What is the order of operations for integers Grade 8? ›First, we solve any operations inside of parentheses or brackets. Second, we solve any exponents. Third, we solve all multiplication and division from left to right. Fourth, we solve all addition and subtraction from left to right.
What is the order of operations for Grade 7 integers? ›The order is PEMDAS: Parentheses, Exponents, Multiplication, and Division (from left to right), Addition and Subtraction (from left to right).
What is the order of operations for 7th grade math? ›
We can remember the order using PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
What are the first 10 integers? ›Hence, first ten intergers are 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 . Q. If is the greatest integer in a group of ten integers, what can you conclude about the other nine integers?
What are the first 4 integers? ›Positive Integers = { 1, 2, 3, 4, ... } Non-Negative Integers = { 0, 1, 2, 3, 4, ... }
What is the sequence 1 2 1 3 10 5 8? ›Fibonacci Numbers (Sequence):
1,1,2,3,5,8,13,21,34,55,89,144,233,377,... Fn=Fn−2+Fn−1 where n≥2 . Each term of the sequence , after the first two, is the sum of the two previous terms. This sequence of numbers was first created by Leonardo Fibonacci in 1202 .
The formula to calculate the sum of integers is given as, S = n(a + l)/2, where, S is sum of the consecutive integers n is number of integers, a is first term and l is last term.
What are the missing number in the sequence 6 11 9 14? ›What are the missing numbers in the sequence 6, 11, _, 9, 14, _? Let's look at the differences between consecutive numbers: 11 - 6 = 5 14 - 9 = 5 The differences are constant, so this is an arithmetic sequence. The common difference is 5. Therefore, the missing numbers are 11 - 5 = 6 and 14 + 5 = 19.
How do you find the integer part of the number? ›The integer part of a real number is the part of the number that appears before the decimal . For example, the integer part of π is 3 , and the integer part of −√2 is −1 .
What is the 2 4 6 8 10 sequence called? ›An arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. 2,4,6,8,10….is an arithmetic sequence with the common difference 2.
What is 2 4 6 8 10 called? ›What is an Even Number? A number that is divisible by 2 and generates a remainder of 0 is called an even number. Examples of even numbers are 2, 4, 6, 8, 10, etc.
What sequence is 1 2 4 8 16 32 64? ›Geometric Sequence
i.e. EX: 1, 2, 4, 8, 16, 32, 64, 128, ...
What is the pattern for 1 4 9 16? ›
Informally: When you multiply an integer (a “whole” number, positive, negative or zero) times itself, the resulting product is called a square number, or a perfect square or simply “a square.” So, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on, are all square numbers.
What is the next number in the series 1 2 2 3 6 7? ›The next number in the series 1, 2, 3, 6, 7, 14, 15, is 33.
What are the sequences 1 4 7 10? ›1, 4, 7, 10, 13, 16, 19, 22, 25, ... This sequence has a difference of 3 between each number.
What are the 3 types of sequence? ›- Arithmetic Sequences.
- Geometric Sequence.
- Fibonacci Sequence.
As a whole number that can be written without a remainder, 0 classifies as an integer.
What is the name of the sequence 1 2 3 4 5? ›He helped spread the use of Hindu systems of writing numbers in Europe (0,1,2,3,4,5 in place of Roman numerals). The seemingly insignificant series of numbers later named the Fibonacci Sequence after him.
What are examples of integer sequences? ›For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description. The sequence 0, 3, 8, 15, ... is formed according to the formula n2 − 1 for the nth term: an explicit definition.
What is the domain of a sequence is the set of ____ numbers? ›A sequence is a function whose domain is the set of natural numbers.
What are the 4 types of integers? ›- Positive Numbers.
- Negative Integer.
To determine the number of integers in between 2 and 8, we can simply write out the numbers and count. Notice that the number of integers in between two other integers will be one less than the range of the two given numbers.
What sequence is 1 1 1 1? ›
Answer and Explanation: Yes, 1 , 1 , 1 , 1 is an arithmetic sequence. It is a trivial example of an arithmetic sequence, but we can see that it satisfies the definition of an arithmetic sequence. This is because, if we take any two consecutive terms of the sequence, they are 1 and 1.
What is the range of a sequence? ›The range is the difference between the smallest and highest numbers in a list or set. To find the range, first put all the numbers in order. Then subtract (take away) the lowest number from the highest.
Is the domain of a sequence the set of positive integers? ›A sequence is a function whose domain is the positive integers. The value of the function at a given integer is a term of the sequence. The range of a sequence is the set of its terms.