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

3,1

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.

Links

Table of n, a(n) for n=3..87.

Github, Dagstuhl's Happy Diner Problem

Example

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 }

Crossrefs

Column 3 gives A318241.

Cf. A107431.

Sequence in context: A236569 A103153 A162022 * A262289 A096918 A075018

Adjacent sequences: A318237 A318238 A318239 * A318241 A318242 A318243

Keyword

nonn,tabl

Author

Floris P. van Doorn, Aug 22 2018

Status

approved