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%I #25 Apr 01 2019 02:12:06
%S 1,1,3,3,5,5,4,4,9,9,11,11,9,9,5,5,12,12,12,12,7,7,23,23,8,8,20,20,29,
%T 29,6,6,33,33,35,35,20,20,39,39,41,41,28,28,12,12,36,36,15,15,51,51,
%U 53,53,36,36,44,44,24,24,20,20,7,7,65,65,36,36,69,69,60,60,42,42,15,15,20,20,52,52,81,81,83,83,9,9,60,60
%N a(1)=1; for n > 1, a(n) = the number of "topped" Mongean shuffles to reorder a stack of n cards to its original order.
%C In the Mongean shuffle, the top card of the stack becomes the top of the new stack, the second of the old stack goes on top of the new stack, the third to the bottom of the new stack, alternating top and bottom of the new stack.
%C Here we define a shuffle where the top-bottom placements in the new stack alternate in the same way, but the second card of the old stack moves to the *bottom* of the stack.
%C A single shuffle is a permutation of 1, 2, 3, 4, 5, 6, ... -> ..., 7, 5, 3, 1, 2, 4, 6, ...
%C The fixed points, where n=a(n), seem to be in A163778.
%C (The "topped" classification is a nomenclature invented here, to be replaced if this variant appears elsewhere in the literature.)
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Shuffling#Mongean_shuffle">Mongean shuffle</a>
%F a(A163778(n)) = A163778(n). - _Andrew Howroyd_, Nov 11 2017
%p topMong := proc(L)
%p ret := [op(1,L)] ;
%p for k from 2 to nops(L) do
%p if type(k,'even') then
%p ret := [op(ret),op(k,L)] ;
%p else
%p ret := [op(k,L),op(ret)] ;
%p end if;
%p end do:
%p ret ;
%p end proc:
%p A238371 := proc(n)
%p local ca,org,tu ;
%p ca := [seq(k,k=1..n)] ;
%p org := [seq(k,k=1..n)] ;
%p for tu from 1 do
%p ca := topMong(ca) ;
%p if ca = org then
%p return tu;
%p end if:
%p end do:
%p end proc:
%p seq(A238371(n),n=2..88) ;
%t topMong[L_] := Module[{ret = {L[[1]]}}, For[k = 2, k <= Length[L], k++, If[ EvenQ[k], ret = Append[ret, L[[k]]], ret = Prepend[ret, L[[k]]]]]; ret];
%t A238371[n_] := Module[{ca, org, tu}, ca = org = Range[n]; For[tu = 1, True, tu++, ca = topMong[ca]; If[ca == org, Return[tu]]]];
%t Array[A238371, 88] (* _Jean-François Alcover_, Jul 03 2018, after _R. J. Mathar_ *)
%o (PARI) apply( A238371(n)=znorder(Mod(bitand(n,2)*2-2,n\2*4+3)), [0..99]) \\ _M. F. Hasler_, Mar 31 2019
%Y Cf. A019567 (Mongean shuffle), A294673 (a bisection).
%K nonn
%O 1,3
%A _R. J. Mathar_, Feb 25 2014
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