A024542 -id:A024542 - cp's OEIS frontend

A024222 Number of shuffles (perfect faro shuffles with cut) required to return a deck of size n to its original order.

0, 1, 2, 2, 4, 4, 3, 3, 6, 6, 10, 10, 12, 12, 4, 4, 8, 8, 18, 18, 6, 6, 11, 11, 20, 20, 18, 18, 28, 28, 5, 5, 10, 10, 12, 12, 36, 36, 12, 12, 20, 20, 14, 14, 12, 12, 23, 23, 21, 21, 8, 8, 52, 52, 20, 20, 18, 18, 58, 58, 60, 60, 6, 6, 12, 12, 66, 66, 22, 22, 35, 35, 9, 9, 20, 20

Offset: 1

Enoch Haga

			a(52)=8: a deck of size 52 returns to its original order in 8 perfect faro shuffles.

Martin Gardner, "Card Shuffles," Mathematical Carnival chapter 10, pp. 123-138. New York: Vintage Books, 1977.
S. Brent Morris, Magic Tricks, Card Shuffling and Dynamic Computer Memories, Math. Assoc. Am., 1998, p. 107.

Tim Folger, Shuffling Into Hyperspace, Discover, 1991 (vol. 12, no. 1), pp. 66-67.
Roger K. W. Hui, Sixteen APL Amuse-Bouches, Vector (2016) Vol. 26, No. 4, 54-66. Art No. 10501480.

A002326 is really the fundamental sequence for this problem. Cf. A024542.

A002326 := proc(n)
    if n =0 then
        1;
    else
        numtheory[order](2,2*n+1) ;
    end if;
end proc:
A024222 := proc(n)
    if n <= 1 then
        n-1 ;
    else
        A002326(floor((n-1)/2)) ;
    end if;
end proc: # R. J. Mathar, Nov 14 2018

A002326 [n_] := If[n == 0, 1, MultiplicativeOrder[2, 2n+1]];
A024222[n_] := If[n <= 1 , n-1, A002326[Floor[(n-1)/2]]];
Table[A024222[n], {n, 1, 76}] (* Jean-François Alcover, May 05 2023, after R. J. Mathar *)

A049206 Maximum mean distance between cards during perfect faro shuffles, with cut, to return to original order in A024222.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 7, 7, 8, 9, 9, 9, 10, 11, 11, 11, 12, 13, 13, 13, 14, 15, 15, 15, 16, 17, 17, 17, 18, 19, 19, 19, 20, 21, 21, 21, 22, 23, 23, 23, 24, 25, 25, 25, 26, 27, 27, 27, 28, 29, 29, 29, 30, 31, 31, 31, 32, 33, 33, 33, 34, 35, 35, 35, 36, 37, 37, 37

Offset: 1

Enoch Haga

			Consider n=6. There are 4 shuffles to return to original order in a 6-card deck. The maximum mean distance between cards during these 4 shuffles and cuts, s1-s4, is 3, computed as follows: s1, 415263, cut, 263415; s2, 421653, cut 653421; s3, 462513, cut 513462; s4, 456123, cut, 123456. Mean distances: s1 15/5=3, maximum; s2 7/5=1.4; s3 13/5=2.6; s4 5/5; mean cumulative distance: 40/20=2.

Cf. A024222, A024542.

Take difference between successive cards after each shuffle. Compute mean (if necessary, round to nearest integer). Retain until replaced by a higher mean in a succeeding shuffle.
(1/4) {2n + 2 - (-1)^[n/2] + (-1)^[(n-1)/2] }. - Ralf Stephan, Jun 10 2005
a(n)=A004525(n), n>1. [From R. J. Mathar, Oct 15 2008]

cp's OEIS Frontend

A024222 Number of shuffles (perfect faro shuffles with cut) required to return a deck of size n to its original order.

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