formula to map between two sequences - math

Question

Given the 3 sequences below, I would like to be able to map seqN to seq0. seq1 maps to seq0 and seq2 to seq1 by:
seqN(i) = (seqN-1(i)*7)%27
For example,
seq1(i) = (seq0(i)*7)%27
seq0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
seq1 0 7 14 21 1 8 15 22 2 9 16 23 3 10 17 24 4 11 18 25 5 12 19 26 6 13 20 27
seq2 0 22 17 12 7 2 24 19 14 9 4 26 21 16 11 6 1 23 18 13 8 3 25 20 15 10 5 27
Now, one way to do this will be to apply the function recursively. But that is not an option for me (this needs to be implemented in hardware). Nor can I save the values of the last sequence to calculate the value of the current sequence - I don't have space for it.
Is there a way to map seqN to seq0 using a mathematical equation (not recursive functions)?

Answer 1

Apart from what #Nitpick pointed out (27 % 27 = 0, so you have either to stop at 26 or use 28 instead), you should simply use a power:
seqN(i) = ( seq0(i) * 7N ) % 27
Or:
seqN(i) = ( seqN-1(i) * 7 ) % 27 = ( seqN-2(i) * 72 ) % 27 = ... = ( seqN-N(i) * 7N ) % 27

Answer 2

Not sure if this helps, but the transition of some seq_N(i) -> seq_/{N+/-1}(i) is completely determined by the following set of cycles:
List(0)
List(1, 7, 22, 19, 25, 13, 10, 16, 4)
List(2, 14, 17, 11, 23, 26, 20, 5, 8)
List(3, 21, 12)
List(6, 15, 24)
List(9)
List(18)
such that 0 -> 0, 1 -> 7 -> 22 -> 19 -> 25 -> ..., etc.
Based on these cycles you could take a number's position and translate it by N % length_of_cycle (just handling cycle wrapping) and get the original number.
I found these cycles with the following code(written in Scala) :
val m = (0 to 26).map(x => (x, x * 7 % 27)).toMap
var cycles = (0 to 26).map(x => {
var x_i = x
val oot = scala.collection.mutable.ArrayBuffer(x)
while (m(x_i) != x)
{
x_i = m(x_i)
oot.append(x_i)
}
oot.toList
})
var filteredCycles = cycles.toArray
for (i <- (1 to 26))
{
val cycle = filteredCycles(i)
for (j <- (1 until cycle.size))
{
filteredCycles(cycle(j)) = List[Int]()
}
}
val uniqueCycles = filteredCycles.filter(_.size != 0)