A301337 Number of steps required in the worst case for two knights to find the princess in a castle with n rooms arranged in a line (Castle and princess puzzle).

1, 1, 2, 2, 2, 3, 4, 4, 6, 6, 6, 7, 8, 8, 10, 10, 10, 11, 12, 12

Offset: 1

Author

Dmitry Kamenetsky, Mar 19 2018

Keywords

• nonn
• more

Comments

The main entry for this problem is A300576. In this version there are two knights who are searching for the princess; each knight can search a different room.

Examples

			For n = 1, there is only room to search, so a(1) = 1.
For n = 2, the knights search both rooms, so a(2) = 1.
For n = 3, the knights can search the first two rooms twice, so a(3) = 2.
For n = 4 and 5, the knights can search the second and the fourth rooms twice, so a(4) = 2 and a(5) = 2.
		

Links

• Dmitry Kamenetsky, Optimal solutions

Crossrefs

Cf. A300576, A301426.

Formula

It seems that for n >= 3:

if n = 0 mod 6, then a(n) = (n/6)*4 - 1,

if n = 1 or 2 mod 6, then a(n) = floor(n/6)*4,

if n = 3, 4 or 5 mod 6, then a(n) = floor(n/6)*4 + 2.

i.e., a(n) = A302404(n-2).