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

1,2

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.

Links

Table of n, a(n) for n=1..14.

Formula

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

Example

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.

Sequence in context: A350457 A010238 A178799 * A089819 A059888 A299204

Adjacent sequences: A360357 A360358 A360359 * A360361 A360362 A360363

Keyword

nonn,more

Author

Pontus von Brömssen, Feb 04 2023

Status

approved