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Let

\mathbb Q_{\geq 0}

be the non-negative rational numbers,

f: \mathbb Q_{\geq 0} \to \mathbb Q_{\geq 0}

such that

f(z+1) = f(z)+1

f(1/z) = f(z)

for

z\neq 0

, and

f(0) = 0.

Define a sequence

P_n

of non-negative integers recursively via P_0 = 0, P_1 = 1, P_n = 2 P_{n-1}+P_{n-2} for every

n \geq 2

. Find

f\left(\frac{P_{20}}{P_{24}}\right).

Proposed by Robert Trosten

Problem Statement