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LinearRecurrence

gives the sequence of length n obtained by iterating the linear recurrence with kernel ker starting with initial values init.

LinearRecurrence

yields terms

through

in the linear recurrence sequence.

MORE INFORMATION

The ker and init can involve arbitrary symbolic expressions, as well as arrays.

The initial list init must be at least as long as the kernel list ker.

If init is longer than ker, only the last Length[ker] elements are used.

LinearRecurrence iterates the recurrence equation with initial conditions , ..., .

When coefficients and initial values are arrays, then the iterated recurrence is interpreted as with dot products of coefficient and values.

If the initial values have dimensions then the coefficients must either be scalar or must have dimensions .

EXAMPLES

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Basic Examples (3)

Solve an initial-value problem for a first-order difference equation with kernel

Find the first few Fibonacci numbers:

In[1]:=

Out[1]=

Solve an initial-value problem for a first-order difference equation with kernel

In[1]:=

Out[1]=

Find the first few Fibonacci numbers:

In[1]:=

Out[1]=

In[2]:=

Out[2]=

Scope (2)

LinearRecurrence works on symbolic kernels and initial values:

LinearRecurrence works on arrays:

Generalizations & Extensions (2)

Generate a subset of values from a given range:

Get only the last value from an iteration:

Applications (1)

Generate recursive sequences, including a Padovan sequence:

Pell numbers:

Pell-Lucas numbers:

Perrin sequence:

Properties & Relations (1)

RSolve finds a symbolic solution for difference equations:

LinearRecurrence generates a procedural solution:

Possible Issues (1)

Initial values are longer than the kernel:

Only the last terms are used:

Neat Examples (1)