Question 2
For each positive integer $n$, let $f(n)$ be the greatest integer for which $2^{f(n)}$ divides
$$a_n=\sum_{i=0}^{\lfloor(n-1)/2\rfloor}\binom n{2i+1}3^i.$$
Find all positive integers $n$ with $f(n)=1996$.
Notation disclosure: the source prints $\binom{2i+1}{n}$, with the selection above the total, consistently with its other reversed binomial notation and its binomial expansion. Here it is explicitly normalized to $\binom n{2i+1}$. Under the usual top-is-total reading of the printed symbol the sum would be zero and $f(n)$ would not be defined.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
$$a_n=\sum_{i=0}^{\lfloor(n-1)/2\rfloor}\binom n{2i+1}3^i.$$
Find all positive integers $n$ with $f(n)=1996$.
Notation disclosure: the source prints $\binom{2i+1}{n}$, with the selection above the total, consistently with its other reversed binomial notation and its binomial expansion. Here it is explicitly normalized to $\binom n{2i+1}$. Under the usual top-is-total reading of the printed symbol the sum would be zero and $f(n)$ would not be defined.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.