A373829 Number of inefficient arrangements in A373182, where inefficient means that the maximum number of persons that a seating arrangement can hold is not achieved.

0, 0, 1, 0, 6, 2, 36, 24, 246, 240, 1920, 2424, 16920, 25920, 166440, 297360, 1809360, 3669840, 21551040, 48666240, 279180720, 691649280, 3908580480, 10501787520, 58813776000, 169809696000, 946627274880, 2914924320000, 16228733875200, 52963370208000

Offset: 1

Author

Enrique Navarrete, Jun 19 2024

Keywords

• nonn

Comments

The maximum number of persons that can be seated in the arrangements in A373182 in n seats is ceiling(n/2).

The seatings here are maximal in the sense that no additional person can be seated without breaking the condition in A373182, but maximum seatings are excluded.

The ratio a(n)/A373182(n) -> 1 as n -> infinity (at a much slower initial rate for even n).

Examples

			a(5)=6 are the following seatings, where _ denotes an empty seat. Seatings of 3 people are the maximum for n=5 and those are not included.
    1 _ _ 2 _
    _ 1 _ 2 _
    _ 1 _ _ 2
    _ 2 _ 1 _
    2 _ _ 1 _
    _ 2 _ _ 1.
For n=9 seats the maximum number of persons that can be seated is 5, hence examples of inefficient arrangements are:
    3 _ 2 _ 1 _ _ 4 _
    _ 3 _ _ 1 _ _ 2 _.
		

Crossrefs

Cf. A081123, A373182.

Formula

a(n) = A373182(n) - (ceiling((n+1)/2))!.