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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 first Hofstadter sequences were described by Douglas Richard Hofstadter in his book Gödel, Escher, Bach.In order of their presentation in chapter III on figures and background (Figure-Figure sequence) and chapter V on recursive structures and processes (remaining sequences), these sequences are:
The order of the sequence is the smallest positive integer such that the sequence satisfies a recurrence of order d, or = for the everywhere-zero sequence. [ citation needed ] The definition above allows eventually- periodic sequences such as 1 , 0 , 0 , 0 , … {\displaystyle 1,0,0,0,\ldots } and 0 , 1 , 0 , 0 , … {\displaystyle 0,1,0,0 ...
In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients [1]: ch. 17 [2]: ch. 10 (also known as a linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear in the various iterates of a variable—that is, in the values of the elements of a sequence.
A sequence () is called hypergeometric if the ratio of two consecutive terms is a rational function in , i.e. (+) / (). This is the case if and only if the sequence is the solution of a first-order recurrence equation with polynomial coefficients.
In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below. They were discovered by mathematician Michael Somos . From the form of their defining recurrence (which involves division), one would expect the terms of the sequence to be fractions, but nevertheless many Somos sequences have ...
Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position.
In the spiral, each triangle shares a side with two others giving a visual proof that the Padovan sequence also satisfies the recurrence relation = + ()Starting from this, the defining recurrence and other recurrences as they are discovered, one can create an infinite number of further recurrences by repeatedly replacing () by () + ()