Search results
Results from the WOW.Com Content Network
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 recurrence of order two satisfied by the Fibonacci numbers is the canonical example of a homogeneous linear recurrence relation with constant coefficients (see below). The Fibonacci sequence is defined using the recurrence
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 ...
A linear recurrence relation expresses the values of a sequence of numbers as a linear combination of earlier values; for instance, the Fibonacci numbers may be defined from the recurrence relation F(n) = F(n − 1) + F(n − 2) together with the initial values F(0) = 0 and F(1) = 1.
In mathematics a P-recursive equation is a linear equation of sequences where the coefficient sequences can be represented as polynomials.P-recursive equations are linear recurrence equations (or linear recurrence relations or linear difference equations) with polynomial coefficients.
There is a general method for expressing the general term of such a sequence as a function of n; see Linear recurrence. In the case of the Fibonacci sequence, one has c 0 = 0 , c 1 = c 2 = 1 , {\displaystyle c_{0}=0,c_{1}=c_{2}=1,} and the resulting function of n is given by Binet's formula .
The Skolem problem is the problem of determining whether a given recurrence sequence has a zero. There exist an algorithm to test whether there are infinitely many zeros, [ 1 ] and if so to find the decomposition of these zeros into periodic sets guaranteed to exist by the Skolem–Mahler–Lech theorem.
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 (,).