A389415 a(n) is the number of permutations of 1..n that win this game. Take a shuffled pack of cards labeled 1..n, repeat this: look at the top card's value, X. Move X cards from the top of the deck to the back, one at a time. If you ever end up with the first card back at the top, you win.

1, 1, 4, 10, 56, 304, 1956, 14856, 130008, 1237440, 13084540, 151656128, 1907139360, 25850635872, 377651957184, 5861540577920, 97217787650368, 1706293248844992, 31680915068666736, 618994402356397440

Offset

1,3

Links

Table of n, a(n) for n=1..20.

Example

For a(3), the permutation 1 2 3 counts as a win. This is what happens:
  1 2 3
  2 3 1 (1 card to the back)
  1 2 3 (2 cards to the back)
  Win!
For a(3), the permutation 1 3 2 is not counted:
  1 3 2
  3 2 1 (1 card to the back)
  3 2 1 (3 cards to the back)
  Infinite loop: lose!
For a(4), the permutation 2 1 3 4 is not counted:
  2 1 3 4
  3 4 2 1
  1 3 4 2
  3 4 2 1
  Infinite loop: lose!

Prog

(Python)
from itertools import permutations
def a(n):
    seq = list(range(1, n+1))
    t = 0
    for p in permutations(seq):
        i = 0
        for step in range(n):
            i += p[i]
            if i == n:
                t += 1
                break
            i = i % n
    return t

Crossrefs

Sequence in context: A225543 A002290 A336997 * A222569 A348514 A254078

Adjacent sequences: A389412 A389413 A389414 * A389416 A389417 A389418

Keyword

nonn,more

Author

Christian Perfect, Oct 03 2025

Extensions

a(12)-a(14) from Michael S. Branicky, Oct 07 2025

a(15)-a(18) from Jason Yuen, Oct 07 2025

a(19)-a(20) from Jason Yuen, Oct 16 2025

Status

approved