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# User:Nicolas Bègue

IT Technician ( ANKAMA GAMES 75 Boulevard d'Armentières 59100 Roubaix FRANCE ) www.facebook.com/nico.begue

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
```

```   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 . ^^