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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.
The equation is called a linear recurrence with constant coefficients of order d. 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]
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.
The Skolem problem is named after Thoralf Skolem, because of his 1933 paper proving the Skolem–Mahler–Lech theorem on the zeros of a sequence satisfying a linear recurrence with constant coefficients. [2] This theorem states that, if such a sequence has zeros, then with finitely many exceptions the positions of the zeros repeat regularly.
Adding a non-linear output mixing function (as in the xoshiro256** and permuted congruential generator constructions) can greatly improve the performance on statistical tests. Another structure for a PRNG is a very simple recurrence function combined with a powerful output mixing function.
The primary difference between a computer algebra system and a traditional calculator is the ability to deal with equations symbolically rather than numerically. The precise uses and capabilities of these systems differ greatly from one system to another, yet their purpose remains the same: manipulation of symbolic equations.
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} .