Padovan Sequence - Padovan L-system

Padovan L-system

If we define the following simple grammar:

variables : A B C

constants : none

start : A

rules : (A → B), (B → C), (C → AB)

then this Lindenmayer system or L-system produces the following sequence of strings:

n = 0 : A

n = 1 : B

n = 2 : C

n = 3 : AB

n = 4 : BC

n = 5 : CAB

n = 6 : ABBC

n = 7 : BCCAB

n = 8 : CABABBC

and if we count the length of each string, we obtain the Padovan sequence of numbers:

1 1 1 2 2 3 4 5 ...

Also, if you count the number of As, Bs and Cs in each string, then for the nth string, you have P(n − 5) As, P(n − 3) Bs and P(n − 4) Cs. The count of BB pairs, AA pairs and CC pairs are also Padovan numbers.