A360360 Given a deck of colored cards, move the top card below the bottom-most card of the same color, with one other card between them. (If the top and bottom cards have the same color, the top card is moved to the bottom of the deck; if there is no other card of the same color, the top card is moved one step down in the deck.) a(n) is the maximum, over all initial color configurations of a deck of n cards, of the length of the eventual cycle when repeatedly applying this move.

1, 2, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64, 64

Offset: 1

Author

Pontus von Brömssen, Feb 04 2023

Keywords

• nonn
• more

Comments

Cards of the same color are considered identical. There can be any number of different colors.

Apparently the lengths of all cycles (not just the longest) are powers of 2.

Examples

			For n = 7, the initial configuration 0120323 (with the top of the deck to the left) leads to a cycle of length 8: 0120323 -> 1203023 -> 2103023 -> 1030232 -> 0130232 -> 1302032 -> 3102032 -> 1020323 -> 0120323. This is the maximum for 7 cards, so a(7) = 8.
		

Crossrefs

Cf. A360361, A360362.

Formula

It appears that a(n) = 2^floor((n-1)/2) for n != 2.