To generate the pattern for order N we first choose a "seed" pair of numbers. This can be any two integers in the range 0 to N - 1. Subsequent values are calculated by iterating
X(n+1) = (X(n) + X(n-1)) Mod N.
Once the "seed" is reencountered the sequence is fated to loop. It is not obvious that the seed is bound to be revisited but I've discovered no counterexamples.
The seed (0,0) always generates the sequence "0, 0, 0, 0, ...". This corresponds to the single point, (0, 0), which appears in every pattern.
The seed "1, 1" starts off "1, 1, 2, 3, 5", except for very low values of N, generating the points (1, 1), (1, 2), (2, 3), (3, 5), .... This sequence always ends with (1, 0). In general, sequences starting (X, X) always end with (X, 0).
The seeds for the third and subsequent sequences are chosen from the set of co-ordinates that haven't been used in previous sequences. The total number of points is always N x N.
To create the pattern, the sets of points generated by each seed are given a different colours.
If in a pattern of order N there is a sequence (a, b, c, ...) then in any pattern whose order is a multiple of N, say NM there will be a sequence (Ma, Mb, Mc, ...) of the same length.
Consequently the patters for composite numbers (like M x N) always contain all the series from their factors (M and N).
If we ignore (0,0) when counting the number if colours then the number of colours required to colour a pattern of composite order will always be more than the number of colours required to colour the patterns for the prime factors.