43
votes
0answers
1k views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where ...
2
votes
2answers
141 views

Finding a closed formula for some SG function

Consider the following sequence, defined by recursion: $g(0)=g(1)=0$. If $n>1$, let $g(n)$ be the mex of the $g(k)$ with $\frac{n}{2} \leq k < n$. The first values of $g(n)$ with $0 \leq n ...
1
vote
1answer
138 views

A strategy in a possibly-infinite game determined by coin toss

A pair of positive integers $x_0$ and $y_0$ are selected and further pairs are generated according to the following rule: After the values of $x_n$ and $y_n$ have been found, a coin is tossed. If it ...