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%I #21 Sep 09 2023 20:49:23
%S 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 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 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 Number of shuffles (perfect faro shuffles with cut) required to return a deck of size n to its original order.
%D Martin Gardner, "Card Shuffles," Mathematical Carnival chapter 10, pp. 123-138. New York: Vintage Books, 1977.
%D S. Brent Morris, Magic Tricks, Card Shuffling and Dynamic Computer Memories, Math. Assoc. Am., 1998, p. 107.
%H Tim Folger, <a href="http://suffe.cool/shuffling.html">Shuffling Into Hyperspace</a>, Discover, 1991 (vol. 12, no. 1), pp. 66-67.
%e a(52)=8: a deck of size 52 returns to its original order in 8 perfect faro shuffles.
%p A002326 := proc(n)
%p if n =0 then
%p 1;
%p else
%p numtheory[order](2,2*n+1) ;
%p end if;
%p end proc:
%p A024222 := proc(n)
%p if n <= 1 then
%p n-1 ;
%p else
%p A002326(floor((n-1)/2)) ;
%p end if;
%p end proc: # _R. J. Mathar_, Nov 14 2018
%t A002326 [n_] := If[n == 0, 1, MultiplicativeOrder[2, 2n+1]];
%t A024222[n_] := If[n <= 1 , n-1, A002326[Floor[(n-1)/2]]];
%t Table[A024222[n], {n, 1, 76}] (* _Jean-François Alcover_, May 05 2023, after _R. J. Mathar_ *)
%Y A002326 is really the fundamental sequence for this problem. Cf. A024542.
%K easy,nonn
%O 1,3
%A _Enoch Haga_
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