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

Author

Enoch Haga

Keywords

• easy
• nonn

Examples

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

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

• 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.

Crossrefs

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

Programs

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 *)