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A318240 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).

%I A318240 #16 Nov 04 2018 23:50:17
%S A318240 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 A318240 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 A318240 13,7,5,5,4,4,3,3,3,3,3,3,15,7,5,5,4,4,3
%N A318240 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 A318240 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 A318240 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 A318240 T(1,k) = 0 and T(n,k) = 1 for 1 < n <= k. These trivial values are omitted in this sequence.
%C A318240 Since every participant can sit with at most (k-1) other participants, T(n,k) >= (n-1)/(k-1).
%C A318240 If A107431(n,k) * (k-1) = n*k - 1 then T(n * k, k) = A107431(n,k).
%C A318240 If A107431(n,k) * (k-1) = n*k - 2 then T(n * k, k) = A107431(n,k) + 1.
%H A318240 Github, <a href="https://github.com/fpvandoorn/Dagstuhl-tables">Dagstuhl's Happy Diner Problem</a>
%e A318240 The triangle begins as follows. The first entry is (n,k) = (3,2).
%e A318240   3
%e A318240   3 3
%e A318240   5 3 3
%e A318240   5 4 3 3
%e A318240   7 4 3 3 3
%e A318240   ...
%e A318240 T(4,2) = 3 from the table assignment { 12/34, 13/24, 14/23 }
%Y A318240 Column 3 gives A318241.
%Y A318240 Cf. A107431.
%K A318240 nonn,tabl
%O A318240 3,1
%A A318240 _Floris P. van Doorn_, Aug 22 2018