User:Nicolas Bègue

Interested in modulo 3 reduction of k-bonacci series and intermediate series , i developed a set of script designed for raspberry pi 3 which calculate the unique set of modulo 3 Pisano periods for each linear recurrence family by iterating over ternary combinations for initializations values : https://github.com/Nico59200/k-bonacci-pisano-finder

Fibonacci/Lucas :

   01120221  length 8

Padovan/Perrin :

   0010111220121  length 13
   0020222110212

tribonacci :

   0011211102021 length 13  
   0022122201012  

tetranacci :

   00011212022012221101020021 length 26 (2*13)
   00022121011021112202010012
   00101211201110020212210222
   1 length 1 
   2

hexanacci :

   one period of length 728 (7*8*13)

For a given linear recurrence family initializations values are numbered according to the linear recurrence formula : Fibonacci : 2 init values Padovan/Perrin/Tribonacci : 3 init values tetranacci : 4 init values pentanacci : 5 init values hexa .... ... So i use a ternary combinations generator to iterate serie calculation over all available combinations corresponding to a given linear recurrence order . e.g :

   ./ternarygen.sh 2
   0 1
   0 2
   1 0
   1 1
   1 2
   2 0
   2 1
   2 2

   ./ternarygen.sh 3
   0 0 1
   0 0 2
   0 1 0
   0 1 1
   0 1 2
   0 2 0
   0 2 1
   0 2 2
   1 0 0
   1 0 1
   1 0 2
   1 1 0
   1 1 1
   1 1 2
   1 2 0
   1 2 1
   1 2 2
   2 0 0
   2 0 1
   2 0 2
   2 1 0
   2 1 1
   2 1 2
   2 2 0
   2 2 1
   2 2 2

once a serie is calculated with a combination of initialization value , it uses a Mathematica wrapper script to find the Pisano period associated . After that a rotating array find unique Periods in the generated list . These periods lengths are Fibonacci numbers or multiples of Fibonacci numbers , some are paired to be complementary to three , the others are auto-complementary to three if splitted in two like the Pisano period 01120221 : 0112 (+) 0221 (=) 0333 .

Feel free to play with my scripts or to implement them in another language . It's open . ^^