Doubling array notation

Doubling array notation is an array notation.

Definition:

An array is a series of nonnegative integers separated by commas. An expression in doubling array notation takes the form of db(_) where an array replaces _. # is any part of the array, including nothing. % is any part of the array that only contains zeroes, including nothing.

db(#,0) = db(#)
db(a) = a+1
db(a,b #) = db(db(a,b-1),b-1 #)
db(a % 0,b #) = db(a % db(a % b-1 #),b-1 #)

Analysis:

db(a) = a+1
db(a,1) = db(db(a)) = a+2
db(a,2) = db(db(a,1),1) = a+4
db(a,3) = db(db(db(a,1),1),1) = a+8
db(a,b) = a+2^b
db(a,0,1) = db(a,db(a)) = a+2^a+1
db(a,1,1) = db(db(a,0,1),0,1) = a+2^a+1+2^{a+2^a+1+1}
db(a,2,1) = db(db(a,1,1),1,1) = a+2^a+1+2^{a+2^a+1+1}+2^{a+2^a+1+2^{a+2^a+1+1}+1}+2^{a+2^a+1+2^{a+2^a+1+1}+2^{a+2^a+1+2^{a+2^a+1+1}+1}+1} -- woah, that's a complicated expression!
from now on, we will use approximations. db(a,2,1) ~ 2^{2^{2^{2^a}}} = E[2]a#4
db(a,3,1) = db(db(a,2,1),2,1) ~ E[2]a#8
db(a,b,1) = f^{2^b}(a) where f(x) = x+2^x ~ E[2]a#(2^b)
db(a,0,2) = db(a,db(a,0,1),1) = db(a,a+2^a+1,1) ~ E[2]a#(2^{a+2^a+1}) ~ 2↑↑2^{a+2^a+1} ~ 2↑↑2^{2^a}
db(a,1,2) = db(db(a,0,2),0,2) ~ 2↑↑2^{2^{2↑↑2^{2^a}}} = 2↑↑(2↑↑2^{2^a}+2)
db(a,2,2) = db(db(a,1,2),1,2) ~ 2↑↑2↑↑2↑↑2↑↑2^{2^a}
db(a,3,2) = db(db(a,1,2),1,2) ~ 2↑↑2↑↑2↑↑2↑↑2↑↑2↑↑2↑↑2↑↑2^{2^a}
db(a,b,2) ~ 2↑↑↑2^b
db(a,0,3) = db(a,db(a,0,2),2) ~ 2↑↑↑2^{2↑↑2^{2^a}} ~ 2↑↑↑2↑↑2^{2^a}
db(a,1,3) = db(db(a,0,3),0,3) ~ 2↑↑↑2↑↑↑2↑↑2^{2^a}
db(a,2,3) = db(db(a,1,3),1,3) ~ 2↑↑↑2↑↑↑2↑↑↑2↑↑↑2↑↑2^{2^a}
db(a,0,4) = db(a,db(a,0,3),3) ~ 2↑↑↑↑2↑↑↑2↑↑2^{2^a}
db(a,0,5) = db(a,db(a,0,3),3) ~ 2↑↑↑↑↑2↑↑↑↑2↑↑↑2↑↑2^{2^a}
db(a,0,0,1) = db(a,0,db(a)) ~ a{a+1}a
db(a,1,0,1) ~ a{a{a+1}a}a
db(a,2,0,1) ~ a{a{a{a{a+1}a}a}a}a
db(a,3,0,1) ~ a{a{a{a{a{a{a{a{a+1}a}a}a}a}a}a}a}a
db(a,0,1,1) = db(a,db(a,0,0,1),0,1) ~ {a,a{a+1}a,1,2}
db(a,1,1,1) = db(db(a,0,1,1),0,1,1) ~ {a,{a,a{a+1}a,1,2},1,2}
db(a,2,1,1) = db(db(a,0,1,1),0,1,1) ~ {a,{a,{a,{a,a{a+1}a,1,2},1,2},1,2},1,2}
db(a,0,2,1) = db(a,db(a,0,1,1),1,1) ~ {a,{a,a{a+1}a,1,2},2,2}
db(a,0,3,1) = db(a,db(a,0,1,1),1,1) ~ {a,{a,{a,a{a+1}a,1,2},2,2},3,2}
db(a,0,0,2) = db(a,0,db(a,0,0,1),1) ~ {a,a,a{a+1}a,2}
db(a,0,0,3) ~ {a,a,{a,a,a{a+1}a,2},3}
db(a,0,0,0,1) ~ {a,a,a,a,a+1}
db(a,0,0,0,0,1) ~ {a,a,a,a,a,a+1}

Alternative definition[]

An array is a sequence of integers of length ω such that there are not infinitely many non-zero entries. The first entry is called the base. The first non-zero entry that is not the base is called the pilot. The value of an array where there are no non-zero entries that are not the base is equal to one more than the base. Otherwise, the value of an array is equal to the value of the array obtained by decreasing the pilot by one, and replacing the entry immediately before the pilot with the value of the array with the pilot decreased by one.