Search results
Results from the WOW.Com Content Network
A famous example is the recurrence for the Fibonacci numbers, = + where the order is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients , because the coefficients of the linear function (1 and 1) are constants that do not depend on n . {\displaystyle n.}
The semi-Fibonacci sequence (sequence A030067 in the OEIS) is defined via the same recursion for odd-indexed terms (+) = + and () =, but for even indices () = (), . The bisection A030068 of odd-indexed terms s ( n ) = a ( 2 n − 1 ) {\displaystyle s(n)=a(2n-1)} therefore verifies s ( n + 1 ) = s ( n ) + a ( n ) {\displaystyle s(n+1)=s(n)+a(n ...
For example, consider the recursive formulation for generating the Fibonacci sequence: F i = F i−1 + F i−2, with base case F 1 = F 2 = 1. Then F 43 = F 42 + F 41, and F 42 = F 41 + F 40. Now F 41 is being solved in the recursive sub-trees of both F 43 as well as F 42. Even though the total number of sub-problems is actually small (only 43 ...
The subproblem of computing F(n − 1) can itself be broken down into a subproblem that involves computing F(n − 2). Therefore, the computation of F(n − 2) is reused, and the Fibonacci sequence thus exhibits overlapping subproblems. A naive recursive approach to such a problem generally fails due to an exponential complexity.
The convergents of the continued fraction for φ are ratios of successive Fibonacci numbers: φ n = F n+1 / F n is the n-th convergent, and the (n + 1)-st convergent can be found from the recurrence relation φ n+1 = 1 + 1 / φ n. [31]
A recursive step — a set of rules that reduces all successive cases toward the base case. For example, the following is a recursive definition of a person's ancestor. One's ancestor is either: One's parent (base case), or; One's parent's ancestor (recursive step). The Fibonacci sequence is another classic example of recursion: Fib(0) = 0 as ...
A classic example of recursion is computing the factorial, which is defined recursively by 0! := 1 and n! := n × (n - 1)!. To recursively compute its result on a given input, a recursive function calls (a copy of) itself with a different ("smaller" in some way) input and uses the result of this call to construct its result.
Multiple recursion can sometimes be converted to single recursion (and, if desired, thence to iteration). For example, while computing the Fibonacci sequence naively entails multiple iteration, as each value requires two previous values, it can be computed by single recursion by passing two successive values as parameters.