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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 .
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.
is constant-recursive because it satisfies the linear recurrence = +: each number in the sequence is the sum of the previous two. [2] Other examples include the power of two sequence ,,,,, …, where each number is the sum of twice the previous number, and the square number sequence ,,,,, ….
A sequence () is called hypergeometric if the ratio of two consecutive terms is a rational function in , i.e. (+) / (). This is the case if and only if the sequence is the solution of a first-order recurrence equation with polynomial coefficients.
A complicated example of a sequence defined by a recurrence relation is Recamán's sequence, [4] defined by the recurrence relation { a n = a n − 1 − n , if the result is positive and not already in the previous terms, a n = a n − 1 + n , otherwise , {\displaystyle {\begin{cases}a_{n}=a_{n-1}-n,\quad {\text{if the result is positive and ...
The first Hofstadter sequences were described by Douglas Richard Hofstadter in his book Gödel, Escher, Bach.In order of their presentation in chapter III on figures and background (Figure-Figure sequence) and chapter V on recursive structures and processes (remaining sequences), these sequences are:
The mom of two was disappointed her district didn't teach phonics as part of its literacy program. She switched her child to a Catholic school where the girl thrived after being taught phonics.
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 ...