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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:
Muller's method is a root-finding algorithm, a numerical method for solving equations of the form f(x) = 0.It was first presented by David E. Muller in 1956.. Muller's method proceeds according to a third-order recurrence relation similar to the second-order recurrence relation of the secant method.
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.
This characterization is because the order-linear recurrence relation can be understood as a proof of linear dependence between the sequences (+) = for =, …,. An extension of this argument shows that the order of the sequence is equal to the dimension of the sequence space generated by ( s n + r ) n = 0 ∞ {\displaystyle (s_{n+r})_{n=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.
The inputs are shown along the bottom of the figure, and the stored element and counter are shown as the symbols and their heights along the black curve. The Boyer–Moore majority vote algorithm is an algorithm for finding the majority of a sequence of elements using linear time and a constant number of words of memory.
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 () + ()