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

%I A024222 #26 Jun 12 2024 10:55:10
%S A024222 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,
%T A024222 28,28,5,5,10,10,12,12,36,36,12,12,20,20,14,14,12,12,23,23,21,21,8,8,
%U A024222 52,52,20,20,18,18,58,58,60,60,6,6,12,12,66,66,22,22,35,35,9,9,20,20
%N A024222 Number of shuffles (perfect faro shuffles with cut) required to return a deck of size n to its original order.
%D A024222 Martin Gardner, "Card Shuffles," Mathematical Carnival chapter 10, pp. 123-138. New York: Vintage Books, 1977.
%D A024222 S. Brent Morris, Magic Tricks, Card Shuffling and Dynamic Computer Memories, Math. Assoc. Am., 1998, p. 107.
%H A024222 Tim Folger, <a href="http://suffe.cool/shuffling.html">Shuffling Into Hyperspace</a>, Discover, 1991 (vol. 12, no. 1), pp. 66-67.
%H A024222 Roger K. W. Hui, <a href="http://archive.vector.org.uk/art10501480">Sixteen APL Amuse-Bouches</a>, Vector (2016) Vol. 26, No. 4, 54-66. Art No. 10501480.
%e A024222 a(52)=8: a deck of size 52 returns to its original order in 8 perfect faro shuffles.
%p A024222 A002326 := proc(n)
%p A024222     if n =0 then
%p A024222         1;
%p A024222     else
%p A024222         numtheory[order](2,2*n+1) ;
%p A024222     end if;
%p A024222 end proc:
%p A024222 A024222 := proc(n)
%p A024222     if n <= 1 then
%p A024222         n-1 ;
%p A024222     else
%p A024222         A002326(floor((n-1)/2)) ;
%p A024222     end if;
%p A024222 end proc: # _R. J. Mathar_, Nov 14 2018
%t A024222 A002326 [n_] := If[n == 0, 1, MultiplicativeOrder[2, 2n+1]];
%t A024222 A024222[n_] := If[n <= 1 , n-1, A002326[Floor[(n-1)/2]]];
%t A024222 Table[A024222[n], {n, 1, 76}] (* _Jean-François Alcover_, May 05 2023, after _R. J. Mathar_ *)
%Y A024222 A002326 is really the fundamental sequence for this problem. Cf. A024542.
%K A024222 easy,nonn
%O A024222 1,3
%A A024222 _Enoch Haga_