%I
%S 3,3,3,5,3,3,5,4,3,3,7,4,3,3,3,7,4,3,3,3,3,9,4,4,3,3,3,3,9,6,4,4,3,3,
%T 3,3,11,6,5,4,3,3,3,3,3,11,6,5,4,3,3,3,3,3,3,13,7,5,5,4,3,3,3,3,3,3,
%U 13,7,5,5,4,4,3,3,3,3,3,3,15,7,5,5,4,4,3
%N Triangle read by rows: T(n,k) = solution to Dagstuhl's Happy Diner Problem with n participants and tables of size at most k (n > k >= 2).
%C There are n participants at a conference, which share meals together in a room with multiple tables. Each table seats at most k participants. T(n,k) is the smallest number of meals so that each participants can share at least one meal with every other participant.
%C There is no requirement on the number of tables, participants can have a meal together more than once, and not every table needs to be fully occupied.
%C T(1,k) = 0 and T(n,k) = 1 for 1 < n <= k. These trivial values are omitted in this sequence.
%C Since every participant can sit with at most (k1) other participants, T(n,k) >= (n1)/(k1).
%C If A107431(n,k) * (k1) = n*k  1 then T(n * k, k) = A107431(n,k).
%C If A107431(n,k) * (k1) = n*k  2 then T(n * k, k) = A107431(n,k) + 1.
%H Github, <a href="https://github.com/fpvandoorn/Dagstuhltables">Dagstuhl's Happy Diner Problem</a>
%e The triangle begins as follows. The first entry is (n,k) = (3,2).
%e 3
%e 3 3
%e 5 3 3
%e 5 4 3 3
%e 7 4 3 3 3
%e ...
%e T(4,2) = 3 from the table assignment { 12/34, 13/24, 14/23 }
%Y Column 3 gives A318241.
%Y Cf. A107431.
%K nonn,tabl
%O 3,1
%A _Floris P. van Doorn_, Aug 22 2018
