Search results
Results from the WOW.Com Content Network
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.
A linear recurrence denotes the evolution of some variable over time, with the current time period or discrete moment in time denoted as t, one period earlier denoted as t − 1, one period later as t + 1, etc. The solution of such an equation is a function of t, and not of any iterate values, giving the value of the iterate at any time.
If the {} and {} are constant and independent of the step index n, then the TTRR is a Linear recurrence with constant coefficients of order 2. Arguably the simplest, and most prominent, example for this case is the Fibonacci sequence , which has constant coefficients a n = b n = 1 {\displaystyle a_{n}=b_{n}=1} .
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 ...
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 ...
It is called recurrent (or persistent) otherwise. [48] For a recurrent state i, the mean hitting time is defined as: = [] = = (). State i is positive recurrent if is finite and null recurrent otherwise. Periodicity, transience, recurrence and positive and null recurrence are class properties — that is, if one state has the property then all ...
For any property partitioning a finite set into finitely many smaller sets, the sum of the cardinalities of the smaller sets equals the cardinality of the bigger set. The Eulerian numbers partition the permutations of n {\displaystyle n} elements, so their sum equals the factorial n ! {\displaystyle n!} .
In mathematics, the Lucas sequences (,) and (,) are certain constant-recursive integer sequences that satisfy the recurrence relation = where and are fixed integers.Any sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences (,) and (,).