Adele knows how to keep us waiting. Rumour Has It, Adele’s fourth studio album will release on November 19.

After her debut album “19,” fans waited three years for her second album, and four years for her third. But for her “30” album release, fans have waited more than six years.

So what can math tell us about this pattern of Adele’s album releases? And can we use this hidden pattern to predict when she’ll release her next grand emotional return?

## A Sequence of Integers

Adele has released four albums in total: “19,” 21,” “25,” and now, “30.”

Yes, these numbers match Adele’s age upon beginning each album. But that’s not the One and Only way to view it. We at AoPS can’t help but see it as an integer sequence

An integer sequence is, quite simply, an ordered list of integers. It can be defined explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms.

For example, we could consider the integer sequence of powers of 2: 1, 2, 4, 8, 16, 32, .... This can be defined explicitly by

$a_n = 2^n$

where

$a_n$

denotes the nth term of the sequence. (Often we think of the first term as the "0th" term.)

A famous integer sequence is the Fibonacci sequence, formed by starting with 0 and 1, and then adding any two consecutive terms to obtain the next term. (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55… ). This can be defined implicitly by

$a_0 = 1, a_1 = 1$

and

$a_n = a_{n-1} + a_{n-2}$

for all

$n \ge 2.$

My favorite sequence is the Catalan numbers. But more on that later.

Seeing Adele's album titles all lined up got us thinking: Did she have some particular integer sequence in mind with her album titles? If so, what is this Adele sequence?

19, 21, 25, 30 …

There are infinite sequences that contain these four terms consecutively. But there are only nine sequences found in the fantastic OEIS.org (On-Line Encyclopedia of Integer Sequences®), an astounding database of almost every conceivable sequence of integers.

OEIS.org was started back in the 1960s by Neil J. A. Sloane, then a graduate student at Cornell University. Today, the OEIS has over 348,000 sequences in its database. If it's a sequence of integers, and someone found it interesting for some reason, then it's almost certainly in the OEIS. Best of all, the OEIS is free for anyone to use!

In the database, you can see the details and continuation of these nine sequences that contain these integers 19, 21, 25, 30 as consecutive terms. Just type in Adele's album numbers, separated by commas, and click "Search."

Three of the nine you’ll see in Math with Bad Drawings. They are A142958 (“roman numerals”), A010412 ("square numbers modulo 51"), and A068442 ("numbers n such that the n-th digit is the same in e and Pi expansions base 3").

Here’s what sequence A142958 (the roman numerals) looks like. (We sure hope we don’t have to wait until she’s 41, though.)

The other six sequences that contain 19, 21, 25, 30 are a bit more complex. Just take a look at A312169! ("Coordination sequence Gal.5.230.1 where G.u.t.v denotes the coordination sequence for a vertex of type v in tiling number t in the Galebach list of u-uniform tilings").

## Predict Adele’s Next Album Release

Which sequence do YOU think Adele is following? Do you think she’s following an identified one, or writing her own?

With any luck, Adele will follow the A072666 sequence (“numbers n such that prime(n) + prime(n+1) - 1 is prime”), and we’ll have a follow-up album by next year.

Or if she follows A068442, we’ll get rapid releases through her 30s.

Let’s just hope it’s not A010412 that she’s following, which has her retiring at 49!

Just as anyone can create a sequence, Adele might just be writing her own — in which case, it’s anyone’s guess!

At AoPS, we teach that you shouldn't assume a pattern continues in the way you think it will, without any evidence.

But if we use past evidence that Adele has tended to follow her own rules — fading into the background for years at a time only to triumphantly reemerge on her watch — we think it’s very possible that she’s writing her own special sequence.

Let’s say Adele's next album turns out to be “34.” Then the sequence that emerges (19, 21, 25, 30, 34) won't match any sequence in the OEIS database. Meaning, we'll have to add it to the database ourselves!

We can only hope they name it the “Adele sequence.”

## OEIS Exploration

You can also search the OEIS by keyword. For example, suppose I wanted to look up my favorite sequence: the Catalan numbers.

I could search the encyclopedia by keyword and bam! I get 4064 results. But the very first one is A000108, the Catalan numbers:

It helpfully tells me how they're defined: the nth Catalan number is

Want to learn more about the Catalan numbers? We cover them in our Intermediate Counting & Probability textbook and online course

OEIS strives to be a thoroughly comprehensive database, so it includes all of the "obvious" sequences that we assume almost everyone knows.

For example, if you search 1, 4, 9, 16, 25, you get as a first result what you probably expect:

But you also get some more obscure results, such as A265055:

Who would have guessed? Again, you shouldn’t assume a sequence continues in the way you might expect!

Finally, a fun feature of the OEIS is that you can “play” sequences as music!

How? The database assigns a note to each number in a sequence, and outputs a music file that you can play.

The OEIS recommends sequence A108618 as one that produces an interesting melody. Just click the “Play" button to give it a listen. (Your browser might download a file that you'll have to play using your computer's media player).

Let's try playing Adele's roman numeral sequence A142958. Just type it into the OEIS "Play a Sequence" page:

Hmmm. Maybe we should leave the music to the professionals like Adele. Hopefully, she won't give us the Cold Shoulder for this!