A019567 Order of the Mongean shuffle permutation of 2n cards: a(n) is least number m for which either 2^m + 1 or 2^m - 1 is divisible by 4n + 1.

1, 2, 3, 6, 4, 6, 10, 14, 5, 18, 10, 12, 21, 26, 9, 30, 6, 22, 9, 30, 27, 8, 11, 10, 24, 50, 12, 18, 14, 12, 55, 50, 7, 18, 34, 46, 14, 74, 24, 26, 33, 20, 78, 86, 29, 90, 18, 18, 48, 98, 33, 10, 45, 70, 15, 24, 60, 38, 29, 78, 12, 84, 41, 110, 8, 84, 26, 134, 12, 46, 35, 36, 68, 146

Offset

0,2

Comments

Write down 1, then 2 to left, 3 to right, 4 to left, ..., getting [ 2n,2n-2,...,4,2,1,3,5,...,2n-1 ]; the sequence 2,3,6,4,6,10,14,5,18,10,12,21,26,9,... gives order of permutation sending 1 to 2n, 2 to 2n-2, ..., 2n to 2n-1.

Equivalently, the sequence 2,3,6,4,6,10,14,5,18,10,12,21,26,9,... gives the number of Mongean shuffles needed to return a deck of 2n cards (n=1,2,3,...) to its original order.

It appears that a(n) = order((-1)^(n+1)*2 in Z_{2n+1}) / f with f=1 when n==2 (mod 3) and for n = 0, 19, 21, 30,33, 52, 55, 61, 63, 70, ..., f=2 else. I don't know how to characterize the "exceptional" n's. - M. F. Hasler, Mar 31 2019

References

A. P. Domoryad, Mathematical Games and Pastimes, Pergamon Press, 1964; see pp. 134-135.

W. W. Rouse Ball, Mathematical Recreations and Essays, 11th ed. 1939, p. 311

Links

R. J. Mathar, Table of n, a(n) for n = 0..2000

P. Diaconis, The mathematics of perfect shuffles, Adv. Appl. Math. 4 (2) (1983) 175-196.

Arne Ledet, The Monge shuffle for two-power decks, Math. Scand. Vol 98, No 1 (2006), 5-11.

E. Ross, Mathematics and Music: The Mathieu Group M_12 (2011), Chapter 2.

T. & X. Vigouroux, First 2000000 terms, for n = 0..1999999

Formula

a(A163777(n)/2) = A163777(n). - Andrew Howroyd, Nov 11 2017

Example

Illustrating the initial terms:
   n  4n+1  2^m+1  2^m-1  m
   0    1            1    1
   1    5     5           2
   2    9     9           3
   3   13    5*13         6
   4   17     17          4
   5   21           3*21  6
   6   25   41*25        10

Maple

A019567:=  proc(n)
    for m from 1 do
        if modp(2^m-1, 4*n+1) =0 or modp(2^m+1, 4*n+1)=0 then
            return m ;
        end if;
    end do;
end proc: # N. J. A. Sloane, Jul 28 2007

Mathematica

a[n_] := For[m=1, True, m++, If[AnyTrue[{-1, 1}, Divisible[2^m+#, 4n+1]&], Return[m]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 26 2019 *)

Prog

(PARI) A019567(n, z=Mod(2, 4*n+1))=for(m=1, oo, bittest(5, lift(z^m+1))&&return(m)) \\ M. F. Hasler, Mar 31 2019

Crossrefs

Cf. A163777, A238371, A294673.

Sequence in context: A349381 A156688 A348243 * A098286 A226615 A138608

Adjacent sequences: A019564 A019565 A019566 * A019568 A019569 A019570

Keyword

nonn,easy

Author

John Bullitt (metta(AT)world.std.com), N. J. A. Sloane and J. H. Conway

Extensions

Comments corrected by Mikko Nieminen, Jul 26 2007, who also provided the Domoryad reference

Definition edited by N. J. A. Sloane, Nov 09 2017

Status

approved