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In mathematics, a Gödel numbering for sequences provides an effective way to represent each finite sequence of natural numbers as a single natural number. While a set theoretical embedding is surely possible, the emphasis is on the effectiveness of the functions manipulating such representations of sequences: the operations on sequences (accessing individual members, concatenation) can be ...
A Gödel numbering can be interpreted as an encoding in which a number is assigned to each symbol of a mathematical notation, after which a sequence of natural numbers can then represent a sequence of symbols. These sequences of natural numbers can again be represented by single natural numbers, facilitating their manipulation in formal ...
A Gödel numbering is a function mapping the set of well-formed formulae (a finite sequence of symbols on which the ordinal notation function is defined) of some formal language to the natural numbers. This associates each well-formed formula with a unique natural number, called its Gödel number.
In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the sequence appear in the equation, for a parameter that is independent of ; this number is called the order of the relation.
The most-common visualization of the Recamán's sequence is simply plotting its values, such as the figure seen here. On January 14, 2018, the Numberphile YouTube channel published a video titled The Slightly Spooky Recamán Sequence , [ 3 ] showing a visualization using alternating semi-circles, as it is shown in the figure at top of this page.
In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger than the one being defined, perhaps even large cardinals (though they can be replaced with recursively large ordinals at the cost of extra technical ...
More generally, every constant-recursive function is asymptotically bounded by an exponential function (see #Closed-form characterization) and the factorial sequence grows faster than this. The Catalan sequence 1 , 1 , 2 , 5 , 14 , 42 , 132 , … {\displaystyle 1,1,2,5,14,42,132,\ldots } is not constant-recursive.
This is in contrast to the definition of sequences of elements as functions of their positions. To define a sequence by recursion, one needs a rule, called recurrence relation to construct each element in terms of the ones before it. In addition, enough initial elements must be provided so that all subsequent elements of the sequence can be ...