A395069 Number of steps to return a deck of n cards to its original state (original order and all cards face up) under a cyclic signed prefix-reversal shuffle: repeatedly flip the top 1 card, then the top 2, ..., then all n cards, cycling back to 1 until the initial state is restored.

2, 8, 9, 24, 50, 72, 28, 64, 162, 60, 121, 240, 234, 392, 75, 160, 204, 648, 228, 400, 294, 264, 529, 504, 200, 1352, 540, 504, 1682, 1800, 186, 384, 2178, 748, 1225, 324, 740, 1140, 1521, 2160, 3362, 336, 1204, 484, 540, 460, 1692, 2304, 1470, 5000, 2601, 624

Offset

1,1

Comments

Starting with a deck of n cards labeled 1..n, all face up, in their natural order from top to bottom, one full step consists of n sub-steps. In sub-step k (k = 1, 2, ..., n), the top k cards are physically reversed as a block: their order is inverted and each card is flipped (face-up becomes face-down and vice versa). This constitutes one full step. The sequence a(n) counts how many full steps are required for the deck to return exactly to the initial state: same order and all cards face up.

Links

Table of n, a(n) for n=1..52.

Fabio Longo, Card Permutation Simulator

Formula

a(n) = n * A002326(n). - Sean A. Irvine, Apr 20 2026

Example

Example for n=3, starting with deck [1u, 2u, 3u]
(u=face up, d=face down, top of deck is leftmost):
Step 1:
  k=1: flip top 1 -> [1d, 2u, 3u]
  k=2: flip top 2 -> [2d, 1u, 3u]
  k=3: flip top 3 -> [3d, 1d, 2u]
Step 2:
  k=1: flip top 1 -> [3u, 1d, 2u]
  k=2: flip top 2 -> [1u, 3d, 2u]
  k=3: flip top 3 -> [2d, 3u, 1d]
Step 3:
  k=1: flip top 1 -> [2u, 3u, 1d]
  k=2: flip top 2 -> [3d, 2d, 1d]
  k=3: flip top 3 -> [1u, 2u, 3u] = initial state
Thus a(3) = 3 full steps = 9 sub-steps.

Prog

(Python)
def a(n):
    # Each card: [id, face] where face 1=up, 0=down
    deck = [[i, 1] for i in range(n)]
    steps = 0
    while True:
        for k in range(1, n + 1):
            packet = deck[:k]
            packet.reverse()
            for card in packet:
                card[1] = 1 - card[1]
            deck = packet + deck[k:]
            steps += 1
            if all(deck[i][0] == i and deck[i][1] == 1 for i in range(n)):
                return steps
print([a(n) for n in range(1, 21)])

Crossrefs

Cf. A002326, A289386, A323712.

Sequence in context: A259672 A163619 A166968 * A075644 A088825 A337706

Adjacent sequences: A395066 A395067 A395068 * A395070 A395071 A395072

Keyword

nonn

Author

Fabio Longo, Apr 10 2026

Extensions

More terms from Sean A. Irvine, Apr 20 2026

Status

approved