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%I #43 Aug 26 2019 05:50:46
%S 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,
%T 12,18,14,12,55,50,7,18,34,46,14,74,24,26,33,20,78,86,29,90,18,18,48,
%U 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
%N 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.
%C 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.
%C 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.
%C 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
%D A. P. Domoryad, Mathematical Games and Pastimes, Pergamon Press, 1964; see pp. 134-135.
%D W. W. Rouse Ball, Mathematical Recreations and Essays, 11th ed. 1939, p. 311
%H R. J. Mathar, <a href="/A019567/b019567.txt">Table of n, a(n) for n = 0..2000</a>
%H P. Diaconis, <a href="http://dx.doi.org/10.1016/0196-8858(83)90009-X">The mathematics of perfect shuffles</a>, Adv. Appl. Math. 4 (2) (1983) 175-196.
%H Arne Ledet, <a href="http://dx.doi.org/10.7146/math.scand.a-14979">The Monge shuffle for two-power decks</a>, Math. Scand. Vol 98, No 1 (2006), 5-11.
%H E. Ross, <a href="http://www.lancaster.ac.uk/pg/rosse2/ProjectYr4EmmaRoss.pdf">Mathematics and Music: The Mathieu Group M_12</a> (2011), Chapter 2.
%H T. & X. Vigouroux, <a href="https://zenodo.org/record/1319615">First 2000000 terms, for n = 0..1999999</a>
%F a(A163777(n)/2) = A163777(n). - _Andrew Howroyd_, Nov 11 2017
%e Illustrating the initial terms:
%e n 4n+1 2^m+1 2^m-1 m
%e 0 1 1 1
%e 1 5 5 2
%e 2 9 9 3
%e 3 13 5*13 6
%e 4 17 17 4
%e 5 21 3*21 6
%e 6 25 41*25 10
%p A019567:= proc(n)
%p for m from 1 do
%p if modp(2^m-1,4*n+1) =0 or modp(2^m+1,4*n+1)=0 then
%p return m ;
%p end if;
%p end do;
%p end proc: # _N. J. A. Sloane_, Jul 28 2007
%t a[n_] := For[m=1, True, m++, If[AnyTrue[{-1, 1}, Divisible[2^m+#, 4n+1]&], Return[m]]];
%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Aug 26 2019 *)
%o (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
%Y Cf. A163777, A238371, A294673.
%K nonn,easy
%O 0,2
%A John Bullitt (metta(AT)world.std.com), _N. J. A. Sloane_ and _J. H. Conway_
%E Comments corrected by Mikko Nieminen, Jul 26 2007, who also provided the Domoryad reference
%E Definition edited by _N. J. A. Sloane_, Nov 09 2017
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