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).

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, 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, 13, 7, 5, 5, 4, 4, 3, 3, 3, 3, 3, 3, 15, 7, 5, 5, 4, 4, 3

Offset: 3

Author

Floris P. van Doorn, Aug 22 2018

Keywords

• nonn
• tabl

Comments

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.

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.

T(1,k) = 0 and T(n,k) = 1 for 1 < n <= k. These trivial values are omitted in this sequence.

Since every participant can sit with at most (k-1) other participants, T(n,k) >= (n-1)/(k-1).

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

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

Examples

			The triangle begins as follows. The first entry is (n,k) = (3,2).
  3
  3 3
  5 3 3
  5 4 3 3
  7 4 3 3 3
  ...
T(4,2) = 3 from the table assignment { 12/34, 13/24, 14/23 }
		

Links

• Github, Dagstuhl's Happy Diner Problem

Crossrefs

Column 3 gives A318241.

Cf. A107431.