

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.


3



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
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OFFSET

0,2


COMMENTS

Write down 1, then 2 to left, 3 to right, 4 to left, ..., getting [ 2n,2n2,...,4,2,1,3,5,...,2n1 ]; 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 2n2, ..., 2n to 2n1.
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.


REFERENCES

A. P. Domoryad, Mathematical Games and Pastimes, Pergamon Press, 1964; see pp. 134135.
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) 175196.
Arne Ledet, The Monge shuffle for twopower decks, Math. Scand. Vol 98, No 1 (2006), 511.
E. Ross, Mathematics and Music: The Mathieu Group M_12 (2011), Chapter 2.


FORMULA

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


EXAMPLE

Illustrating the initial terms:
n 4n+1 2^m+1 2^m1 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^m1, 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


CROSSREFS

Cf. A163777, A238371, A294673.
Sequence in context: A209775 A125703 A156688 * A098286 A226615 A138608
Adjacent sequences: A019564 A019565 A019566 * A019568 A019569 A019570


KEYWORD

nonn,easy,changed


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



