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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.
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.
Then the recurrence relation is used to successively compute trial values for , down to . Noting that a second sequence obtained from the trial sequence by multiplication by a constant normalizing factor will still satisfy the same recurrence relation, one can then apply a separate normalizing relationship to determine the normalizing factor ...
Single-step methods (such as Euler's method) refer to only one previous point and its derivative to determine the current value. Methods such as Runge–Kutta take some intermediate steps (for example, a half-step) to obtain a higher order method, but then discard all previous information before taking a second step. Multistep methods attempt ...
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 ...
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
Also, the Motzkin number for n gives the number of routes on the upper right quadrant of a grid from coordinate (0, 0) to coordinate (n, 0) in n steps if one is allowed to move only to the right (up, down or straight) at each step but forbidden from dipping below the y = 0 axis.
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 ...