Farey sequences

Returns the Farey sequence of order \(n\)

farey(n, print=FALSE, give.series = FALSE)

Arguments

n

Order of Farey sequence

print

Boolean, with TRUE meaning to print out the text version of the Farey sequence in human-readable form. Default value of FALSE means not to print anything

give.series

Boolean, with TRUE meaning to return the series explicitly, and default FALSE meaning to return a 3 dimensional array as detailed below

Details

If give.series takes its default value of FALSE, return a matrix a of dimension c(2,u) where u is a (complicated) function of n. If v <- a[i,], then v[1]/v[2] is the \(i^{\mathrm{th}}\) term of the Farey sequence. Note that det(a[(n):(n+1),])== -1

If give.series is TRUE, then return a matrix a of size c(4,u-1). If v <- a[i,], then v[1]/v[2] and v[3]/v[4] are successive pairs of the Farey sequence. Note that det(matrix(a[,i],2,2))== -1

References

G. H. Hardy and E. M. Wright 1985. An introduction to the theory of numbers, Oxford University Press (fifth edition)

Author

Robin K. S. Hankin

See also

unimodular

Examples

farey(3)
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    1    1    2
#> [2,]    1    3    2    3
#> [3,]    1    1    2    1
#> [4,]    3    2    3    1