Question:

What comes next in the sequence 1, 11, 21, 1211, 111221?

Answer:

Each number describes the previous one. 11 means "one 1", 1211 means "one 2, one 1". So, 312211 is next, "3 1's, 2 2's, 1 1".

More Info:

Mathematics
Integer sequences

In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers.

An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms. 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.

In mathematics, the Kolakoski sequence (named after William Kolakoski, who discussed it in 1965) is an infinite sequence of symbols {1,2} which is its own run-length encoding.

The initial terms of the sequence are:


Sequence learning

Sequence learning is inherent to human ability because it is an integrated part of conscious and nonconscious learning as well as activities. Sequences of information or sequences of actions are used in various everyday tasks: "from sequencing sounds in speech, to sequencing movements in typing or playing instruments, to sequencing actions in driving an automobile." Sequence learning can be used to study skill acquisition and in studies of various groups ranging from neuropsychological patients to infants. According to Ritter and Nerb, “The order in which material is presented can strongly influence what is learned, how fast performance increases, and sometimes even whether the material is learned at all.” Sequence learning, more known and understood as a form of explicit learning, is now also being studied as a form of implicit learning as well as other forms of learning. Sequence learning can also be referred to as sequential behavior, behavior sequencing, and serial order in behavior.

Entertainment is a form of activity that holds the attention and interest of an audience, or gives pleasure and delight. It can be an idea or a task, but is more likely to be one of the activities or events that have developed over thousands of years specifically for the purpose of keeping an audience's attention. Although people's attention is held by different things, because individuals have different preferences in entertainment, most forms are recognisable and familiar. Storytelling, music, drama, dance, and different kinds of performance exist in all cultures, were supported in royal courts, developed into sophisticated forms and over time became available to all citizens. The process has been accelerated in modern times by an entertainment industry which records and sells entertainment products. Entertainment evolves and can be adapted to suit any scale, ranging from an individual who chooses a private entertainment from a now enormous array of pre-recorded products; to a banquet adapted for two; to any size or type of party, with appropriate music and dance; to performances intended for thousands; and even for a global audience.

The experience of being entertained has come to be strongly associated with amusement, so that one common understanding of the idea is fun and laughter, although many entertainments have a serious purpose. This may be the case in the various forms of ceremony, celebration, religious festival, or satire for example. Hence, there is the possibility that what appears as entertainment may also be a means of achieving insight or intellectual growth.

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