A360362 -id:A360362 - cp's OEIS frontend

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

Pontus von Brömssen, Feb 04 2023

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.

			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.

Cf. A360361, A360362.

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

A360361 Maximum length of the transient part when repeatedly applying the move described in A360360 to a deck of n colored cards.

Original entry on oeis.org

0, 0, 1, 4, 6, 11, 18, 24, 44, 58, 96, 120, 210, 254

Offset: 1

Pontus von Brömssen, Feb 04 2023

			For n = 5, the initial configuration 01112 (with the top of the deck to the left) requires 6 moves to reach the first configuration in the eventual cycle: 01112 -> 10112 -> 01121 -> 10121 -> 01211 -> 10211 -> 02111 -> 20111 -> 02111. This is the maximum for 5 cards, so a(5) = 6.
Lexicographically first optimal initial configuration for 1 <= n <= 14:
   n  a(n)  configuration
   1    0   0
   2    0   00
   3    1   001
   4    4   0012
   5    6   01112
   6   11   011023
   7   18   0111023
   8   24   01221034
   9   44   012110234
  10   58   0111234234
  11   96   01200321345
  12  120   011102345345
  13  210   0122112345345
  14  254   01112345326546

Cf. A360360, A360362.

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