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##Haskell, 43 bytes

q=((!!).).iterate
g=q(`q`1)(3*)
q(`g`3)4$64

There has be a better way to flip g inline.

46 bytes:

i=iterate
n%0=3*n
n%m=i(%(m-1))1!!n
i(3%)4!!64

48 bytes:

n%1=3^n
1%m=3
n%m=(n-1)%m%(m-1)
iterate(3%)4!!64

Just writing down the definitions.

The base cases are a bit cleaner backed up to 0, though it saves no bytes. Perhaps it will make it easier to write an alternate definition.

n%0=3*n
0%m=1
n%m=(n-1)%m%(m-1)
z=iterate(3%)2!!1

Haskell, 43 bytes

q=((!!).).iterate
g=q(`q`1)(3*)
q(`g`3)4$64

There has be a better way to flip g inline.

46 bytes:

i=iterate
n%0=3*n
n%m=i(%(m-1))1!!n
i(3%)4!!64

48 bytes:

n%1=3^n
1%m=3
n%m=(n-1)%m%(m-1)
iterate(3%)4!!64

Just writing down the definitions.

The base cases are a bit cleaner backed up to 0, though it saves no bytes. Perhaps it will make it easier to write an alternate definition.

n%0=3*n
0%m=1
n%m=(n-1)%m%(m-1)
z=iterate(3%)2!!1

##Haskell, 43 bytes

q=((!!).).iterate
g=q(`q`1)(3*)
q(3`g`)4$64

There has be a better way to flip g inline.

45 bytes:

i=iterate
n%0=3*n
n%m=i(%(m-1))1!!n
i(3%)4!!64

48 bytes:

n%1=3^n
1%m=3
n%m=(n-1)%m%(m-1)
iterate(3%)4!!64

Just writing down the definitions.

The base cases are a bit cleaner backed up to 0, though it saves no bytes. Perhaps it will make it easier to write an alternate definition.

n%0=3*n
0%m=1
n%m=(n-1)%m%(m-1)
z=iterate(3%)2!!1

##Haskell, 43 bytes

q=((!!).).iterate
g=q(`q`1)(3*)
q(`g`3)4$64

There has be a better way to flip g inline.

46 bytes:

i=iterate
n%0=3*n
n%m=i(%(m-1))1!!n
i(3%)4!!64

48 bytes:

n%1=3^n
1%m=3
n%m=(n-1)%m%(m-1)
iterate(3%)4!!64

Just writing down the definitions.

The base cases are a bit cleaner backed up to 0, though it saves no bytes. Perhaps it will make it easier to write an alternate definition.

n%0=3*n
0%m=1
n%m=(n-1)%m%(m-1)
z=iterate(3%)2!!1

##Haskell, 45 bytes

i=iterate
n%0=3*n
n%m=i(%(m-1))1!!n
i(3%)4!!64

48 bytes:

n%1=3^n
1%m=3
n%m=(n-1)%m%(m-1)
iterate(3%)4!!64

Just writing down the definitions.

The base cases are a bit cleaner backed up to 0, though it saves no bytes. Perhaps it will make it easier to write an alternate definition.

n%0=3*n
0%m=1
n%m=(n-1)%m%(m-1)
z=iterate(3%)2!!1

##Haskell, 43 bytes

q=((!!).).iterate
g=q(`q`1)(3*)
q(3`g`)4$64

There has be a better way to flip g inline.

45 bytes:

i=iterate
n%0=3*n
n%m=i(%(m-1))1!!n
i(3%)4!!64

48 bytes:

n%1=3^n
1%m=3
n%m=(n-1)%m%(m-1)
iterate(3%)4!!64

Just writing down the definitions.

The base cases are a bit cleaner backed up to 0, though it saves no bytes. Perhaps it will make it easier to write an alternate definition.

n%0=3*n
0%m=1
n%m=(n-1)%m%(m-1)
z=iterate(3%)2!!1