matrixcalc (version 1.0-3)

fibonacci.matrix: Fibonacci Matrix

Description

This function constructs the order n + 1 square Fibonacci matrix which is derived from a Fibonacci sequence.

Usage

fibonacci.matrix(n)

Arguments

n

a positive integer value

Value

An order \(n + 1\) matrix

Details

Let \(\left\{ {{f_0},\;{f_1},\; \ldots ,\;{f_n}} \right\}\) be the set of \( n + 1\) Fibonacci numbers where \({f_0} = {f_1} = 1\) and \({f_j} = {f_{j - 1}} + {f_{j - 2}},\quad 2 \le j \le n\). The order \(n + 1\) Fibonacci matrix \({\bf{F}}\) has as typical element \({F_{i,j}} = \left\{ {\begin{array}{*{20}{c}} {{f_{i - j + 1}}}&{i - j + 1 \ge 0}\\ 0&{i - j + 1 < 0} \end{array}} \right.\).

References

Zhang, Z. and J. Wang (2006). Bernoulli matrix and its algebraic properties, Discrete Applied Nathematics, 154, 1622-1632.

Examples

# NOT RUN {
F <- fibonacci.matrix( 10 )
print( F )
# }