I first became interested in this problem when my brother, in secondary school, was asked to determine the nth term of a particularly challenging sequence, got stuck, and showed it to me. The question stated that there were $n$ weather stations, each with their own piece of data that they had collected, and each weather station had to know every piece of data. The question stated that they were to use phone calls to transfer the data, so that the two weather stations in the call could synchronise the information they had. The sequence was to be generated from the fewest number of calls needed to synchronise the data of $n$ weather stations.
For example, if A calls B, now both A and B have data AB. If B were to then call C, A would have AB, B would have ABC and C ABC likewise. I quickly determined that the upper bound of this sequence would be $n!$, which dropped to $n^2$ with further thought. Edit: The upper bound of calls needed has dropped to $(n-1)^2$ with help from Mindlack. However, I couldn't help my brother with his homework (of GCSE standard) so we dropped the matter.
I maintained interest, so I generated this table of values, $c$ and $n$, which is correct to the best of my ability: \begin{array} {|r|r|}\hline n & c \\ \hline 0 & undefined \\ \hline 1 & 0 \\ \hline 2 & 1 \\ \hline 3 & 3 \\ \hline 4 & 4 \\ \hline 5 & 8 \\ \hline 6 & 10 \\ \hline 7 & 11 \\ \hline 8 & 12 \\ \hline \end{array}
I noticed that the powers of two followed the rule $c = $ $0.5$ $n$ $log_2(n)$, but I still had no rule for finding the other terms.I reached out to my maths teacher (A level), and we went over it together. He was totally stumped however, and I was having trouble hand generating efficient solutions for higher values of $n$.
I am almost positive that this sequence must have been generated before, especially in the information theory branches of mathematics, as the idea of sharing data over a two way connection would be well researched. When I googled it, I only found reference to $0.5$ $n$ $log_2(n)$ in matrix multiplication and computational complexity.
Could someone help me find the elusive $n$th term formula for this sequence, or point me to where this sequence has been generated before?