A035485 Card on top of deck at n-th stage of R. K. Guy's shuffling problem.

1, 2, 3, 1, 6, 5, 9, 1, 4, 2, 16, 10, 12, 14, 23, 16, 18, 20, 17, 27, 30, 33, 38, 10, 14, 37, 32, 6, 11, 19, 53, 37, 25, 21, 12, 34, 38, 8, 50, 48, 46, 14, 18, 23, 47, 53, 84, 52, 31, 49, 1, 51, 91, 61, 42, 79, 4, 29, 6, 49, 26, 23, 115, 4, 70, 93, 109, 11, 16, 19, 49, 18, 124, 97, 70, 10, 134, 111, 7, 38, 14, 79, 11, 129

Offset

0,2

Comments

At n-th step, pick up top n cards and interlace them with the next n.

Here is the deck after steps 0,1,2,3,4,5:

1,2,3,4,5,6,7,...

2,1,3,4,5,6,7,...

3,2,4,1,5,6,7,...

1,3,5,2,6,4,7,8,9,...

6,1,4,3,7,5,8,2,9,10,...

It is conjectured that eventually every number appears on top of the deck.

See A035491 for (the relevant part of) the deck after the n-th step. - M. F. Hasler, Aug 13 2022

References

D. Gale, Mathematical Entertainments: "Careful Card-Shuffling and Cutting Can Create Chaos," The Mathematical Intelligencer, vol. 14, no. 1, 1992, pages 54-56.

D. Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998.

Links

Lars Blomberg, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Perfect Shuffle.

Formula

a(n) = A035491(n,1), i.e., the first element of the n-th row of that table, for all n > 0. - M. F. Hasler, Aug 13 2022

Prog

(Python)
def aupton(terms):
  alst, deck = [1], list(range(1, 2*terms+1))
  for n in range(1, terms+1):
    first, next = deck[:n], deck[n:2*n]
    deck[0:2*n:2] = next
    deck[1:2*n:2] = first
    alst.append(deck[0])
  return alst
print(aupton(83)) # Michael S. Branicky, Feb 01 2021
(PARI) A035485(n)=A035491_row(n+!n)[1]-!n \\ M. F. Hasler, Aug 13 2022

Crossrefs

See A035491 for the array, also A035490, A035492.

Sequence in context: A110237 A189970 A076631 * A074306 A294218 A356013

Adjacent sequences: A035482 A035483 A035484 * A035486 A035487 A035488

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Clark Kimberling

Extensions

More terms from Jud McCranie

Status

approved