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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). 1

%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 (k-1) other participants, T(n,k) >= (n-1)/(k-1).

%C If A107431(n,k) * (k-1) = n*k - 1 then T(n * k, k) = A107431(n,k).

%C If A107431(n,k) * (k-1) = n*k - 2 then T(n * k, k) = A107431(n,k) + 1.

%H Github, <a href="https://github.com/fpvandoorn/Dagstuhl-tables">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

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Last modified August 12 21:10 EDT 2020. Contains 336440 sequences. (Running on oeis4.)