The function
is defined on the set of nonnegative integers and takes nonnegative integer values satisfying the following conditions: for every
(i)
(ii)
(iii)
Prove that for each positive integer
the number of integers
with
and
is
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The function $F$ is defined on the set of nonnegative integers and takes nonnegative integer values satisfying the following conditions: for every $n \geq 0,$
(i) $F(4n) = F(2n) + F(n),$
(ii) $F(4n + 2) = F(4n) + 1,$
(iii) $F(2n + 1) = F(2n) + 1.$
Prove that for each positive integer $m,$ the number of integers $n$ with $0 \leq n < 2^m$ and $F(4n) = F(3n)$ is $F(2^{m + 1}).$
Školjka