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Jelly, 13 bytes

æḟ©3Ḥạ++_®)»ċ

A monadic Link that accepts a non-negative integer, \$N\$, and yields the value of \$f(n)\$.

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How?

Outputs the number of terms less than or equal to \$N\$ in another sequence - the unique, strictly increasing sequence of non-negative integers, \$a\$, such that \$a(a(n)) = 3n\$. This is A003605, which gives the indices at which an increase occurs in \$f\$.

æḟ©3Ḥạ++_®)»ċ - Link: non-negative integer, N
 ) - for each {I in [1..N] ([] if N==0)}:
æḟ 3 - floor to the nearest power of three
 © - (and copy this to the register)
 Ḥ - double {that}
 ạ - {that} absolute difference {I}
 + - do this twice:
 + - add {I}
 _® - subtract the value stored in the register
 » - maximum with {N} (vectorises)
 ċ - count occurrences of {N}