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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 .
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} .
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 ,,,,, ….
Single-step methods (such as Euler's method) refer to only one previous point and its derivative to determine the current value. Methods such as Runge–Kutta take some intermediate steps (for example, a half-step) to obtain a higher order method, but then discard all previous information before taking a second step. Multistep methods attempt ...
For example, C to D (major second) is a step, whereas C to E (major third) is a skip. More generally, a step is a smaller or narrower interval in a musical line, and a skip is a wider or larger interval with the categorization of intervals into steps and skips is determined by the tuning system and the pitch space used.
One class of examples is the staggered geometric progressions that get closer to their limits only every other step or every several steps, for instance the example () =,, /, /, /, /, …, / ⌊ ⌋, … detailed below (where ⌊ ⌋ is the floor function applied to ). The defining Q-linear convergence limits do not exist for this sequence ...
The number of ways of writing n as an ordered sum in which each term is odd and greater than 1 is equal to P(n − 5). For example, P(6) = 4, and there are 4 ways to write 11 as an ordered sum in which each term is odd and greater than 1: 11 ; 5 + 3 + 3 ; 3 + 5 + 3 ; 3 + 3 + 5