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

1,3

References

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.

Links

Table of n, a(n) for n=1..76.

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.

Example

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

Maple

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

Mathematica

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 *)

Crossrefs

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

Sequence in context: A263856 A090277 A324662 * A196063 A205450 A215674

Adjacent sequences: A024219 A024220 A024221 * A024223 A024224 A024225

Keyword

easy,nonn

Author

Enoch Haga

Status

approved